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Optimization Models for Capacity Planning in Health Care Delivery

Permanent Link: http://ufdc.ufl.edu/UFE0021583/00001

Material Information

Title: Optimization Models for Capacity Planning in Health Care Delivery
Physical Description: 1 online resource (125 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: capacity, healthcare, metaheuristic, optimization, queueing
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Health care capacity planning is the art and science of predicting the quantity of resources required to deliver health care service at specified levels of cost and quality. Because of variability in the arrival of patients and in the delivery of health care services, successfully meeting the demand for health care services is a daunting task that requires an understanding of the inherent trade-off between its cost and quality of service. In our work, we model the general health care systems as queueing stations and incorporate queueing theory into an optimization framework. The queueing modeling approach captures the stochastic nature of arrivals and service times that is typical in health care systems. The optimization framework determines the minimum cost capacity (e.g., health care teams, beds) required to achieve a target level of customer service. The queueing equations lead to nonlinear equations in the model. In addition, for the types of health care organizations we are considering, capacity is typically available only in discrete options (e.g., number of employees or beds). Thus, the capacity planning models of interest are nonlinear optimization problems with integer variables. We develop effective solution algorithms to obtain high quality solutions particularly for realistic-sized problems. For the analysis of underlying queuing systems, we either use available results from the literature or develop approximations. For the solution of optimization models, we employ network optimization, meta-heuristic, and Lagrangian relaxation approaches to develop effective solution algorithms. We present results from extensive computational experiments to demonstrate the computational efficiency and effectiveness of the proposed solution approaches.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Akcali, Elif.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0021583:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021583/00001

Material Information

Title: Optimization Models for Capacity Planning in Health Care Delivery
Physical Description: 1 online resource (125 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: capacity, healthcare, metaheuristic, optimization, queueing
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Health care capacity planning is the art and science of predicting the quantity of resources required to deliver health care service at specified levels of cost and quality. Because of variability in the arrival of patients and in the delivery of health care services, successfully meeting the demand for health care services is a daunting task that requires an understanding of the inherent trade-off between its cost and quality of service. In our work, we model the general health care systems as queueing stations and incorporate queueing theory into an optimization framework. The queueing modeling approach captures the stochastic nature of arrivals and service times that is typical in health care systems. The optimization framework determines the minimum cost capacity (e.g., health care teams, beds) required to achieve a target level of customer service. The queueing equations lead to nonlinear equations in the model. In addition, for the types of health care organizations we are considering, capacity is typically available only in discrete options (e.g., number of employees or beds). Thus, the capacity planning models of interest are nonlinear optimization problems with integer variables. We develop effective solution algorithms to obtain high quality solutions particularly for realistic-sized problems. For the analysis of underlying queuing systems, we either use available results from the literature or develop approximations. For the solution of optimization models, we employ network optimization, meta-heuristic, and Lagrangian relaxation approaches to develop effective solution algorithms. We present results from extensive computational experiments to demonstrate the computational efficiency and effectiveness of the proposed solution approaches.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Akcali, Elif.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0021583:00001


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1b703f05458f4e49481a40d7bf28f8112fc208d3







OPTIMIZATION MODELS FOR CAPACITY PLANNING IN HEALTH CARE
DELIVERY



















By
CHIN-I LIN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008


































S2008 Chin-I Lin


































To my family










ACKENOWLED GMENTS

I would like to thank all people who have helped and inspired me during my doctoral

study. I want to express my sincere gratitude to my dissertation advisor, Dr. Elif Akgahl,

for her guidance, insight and support during this research and study. I am also grateful

to my coninittee niembers, Dr. Farid AitSahlia, Dr. P. Oscar Boykin and Dr. Siriphong

L leph.- ii" ol..IsI ch, for their constructive -II_a-r;-- -Hun and coninents. I wish to extend my

warmest thanks to my mentor, Dr. Shangyao Yan, for leading me to the field of operations

research, his friendship and numerous fruitful discussions. My deepest gratitude goes to

to my parents, Zhe-Xiong and Fang-Xue, and my husband, Chlungt-Jui. Without their

understanding and encouragement, it would have been impossible for me to complete my

degree.










TABLE OF CONTENTS


page


ACKNOWLEDGMENTS

LIST OF TABLES.

LIST OF FIGURES

ABSTRACT

CHAPTER

1 INTRODUCTION

2 AGGREGATE HOSPITAL BED CAPACITY PLANNING .......


2.1 Introduction.
2.2 Literature Review.
2.3 Problem Formulation.
2.3.1 Restricted Bed Capacity Planning Problem.
2.3.2 Restricted Bed Capacity Planning Problem with Shu
2.4 Illustration of the Model.
2.4.1 A Representative Decision-Making Scenario.
2.4.2 Experiment 1 -An Application of the Model.
2.4.3 Experiment 2 -Assessing the Impact of Problem Pa
2.5 Extensions.
2.6 Concluding Remarks and Future Research Directions

3 HEALTH CARE TEAM CAPACITY PLANNING

3.1 Introduction.
3.2 Literature Review.
3.3 Problem Formulation.
3.4 Queueing Analysis
3.4.1 Preemptive Case: Emergency Medicine Services
3.4.2 Non-Preemptive Case: Outpatient Clinic Services
3.5 Computational Study.
3.5.1 Computational Performance of the DBA Method.
3.5.2 Computational Performance of the HCTSCP Model
3.6 Concluding Remarks and Future Research Directions

4 HOSPITAL BED ALLOCATION PROBLEM.

4.1 Introduction.
4.2 Literature Review.
4.3 Problem Formulation.
4.4 Solution Algorithms


Ittering .


rameters











4.4.1 Genetic Algorithm . ....... .... 75
4.4.2 Greedy Randomized Adaptive Search Procedure .. .. .. .. 80
4.4.3 Hybridization of GA & GRASP ..... .. . 81
4.5 Computational Study ........ . .. .. 83
4.5.1 Summary of Results Obtained by GA ... ... .. .. 85
4.5.2 Summary of Results Obtained by GRASP ... .. .. 88
4.5.3 Summary of Results Obtained by HA ... .. .. 88
4.6 Concluding Remarks and Future Research Directions .. .. .. 92

5 EMERGENCY ROOM SERVICES FACILITY LOCATION AND CAPACITY
PLANNING ... ......... ........... 94

5.1 Introduction ......... . .. .. 94
5.2 Literature Review ......... .. .. 95
5.3 Problem Formulation ......... .. .. 97
5.4 Solution Approach ......... .. .. .. 100
5.4.1 Lagrangian Relaxation Approach .... ... .. 100
5.4.2 Lower Bound ......... .. .. 101
5.4.3 Upper Bound ......... .. .. 103
5.4.4 Lagfrangfian Multipliers . ...... .. 106
5.5 Computational Study ......... .. .. 107
5.5.1 Experimental Desigfn . .. .. 107
5.5.2 Experimental Results .. . .. 108
5.6 Concluding Remarks and Future Research Directions .. .. .. 111

6 CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS .. .. .. .. 114

REFERENCES ......... . .. . 117

BIOGRAPHICAL SK(ETCH ....._._. .. .. 125











LIST OF TABLES


page

27

27



:31



:32


Table

2-1 Parameter settings for the base scenario, S1 .........

2-2 Scenario descriptions for experiment 1 .........

2-3 Suninary statistics for the RBCPwS problent's solution time (in CPU seconds)
as a function of initial effective capacity .........

2-4 Suninary statistics for the RBCPwS problent's solution as a function of capacity
levels and the length of the planning horizon ..........


The possible transitions enter state (ni, n2,m) for the ED setting ....

Computational requirement of the DBA, AGE, and GE methods .....

Relative and percentage error: preeniptive case ...

Relative and percentage error: non-preeniptive case ....

Team configurations ......

Parameter settings ......

Impact of the fraction of class 1 patients on the similarity index .....

Impact of unit patient delay cost on the similarity index ..... .

Impact of nmaxiniun allowable average time in system on the similarity in

Near-optinmal solutions obtained by using CPLEX .......

Solutions obtained by GA ......

Solutions obtained by GR ASP .......

Solutions obtained by HA ......

Experimental factor settings .......

Experiment 1: effects of nmaxiniun number of facilities opened ......

Experiment 2: effects of capacity setting ......

Experiment :3: effects of diversion probability ......

Experiment 4: effects of time value ......

Performance of the heuristic ...


. 49

. 58

. 60

. 61

. 6:3

. 6:3

. 65

. 65

dex .. 65

. 86

.. 87

. 89

. 91

.. 107

.. 109

.... 110

.... 110

.. 111

S112











LIST OF FIGURES


Figure

2-1 Network flow representation for RBCP with co=300, B=25, n=1, and T=4


page

21


2-2

2-3

2-4

2-5

3-1

3-2

4-1

4-2

4-3

4-4

4-5

4-6

4-7

4-8

4-9

4-10

4-11

4-12

4-13

5-1

5-2

5-3

5-4

5-5


Network flow representation for RBCPwS with co=275, B=25,

Patient arrival rate for experiment 1 .....

Optimal capacity plans for experiment 1 .....

Number of nodes in the network as a function of initial effective

An illustration of the network representation for HCTSCP ..

Two-dimensional CT3iC approximation .....

Pseudo-code of the genetic algorithm ......

Pseudo-code of the population generating procedure ......

Pseudo-code of occupancy-driven allocation .....

Pseudo-code of random-rectified procedure ...

Pseudo-code of crossover procedure ...

Example of crossover ......

Pseudo-code of the mutation procedure ......

Pseudo-code of GRASP .....

Pseudo-code of greedy randomized construction procedure ..

Pseudo-code of local search procedure ...

Pseudo-code of HA ......

Pseudo-code of elite set generation procedure .....

The updating functions of a ...

Pseudo-code of the Lagrangian relaxation .....

Pseudo-code of the feasible solution generation ......

Pseudo-code of covering all demand nodes .....

Pseudo-code of closing facilities .....

Convergence speed of the modified LR .....


n=1, and T=4


. 28

. 28

e bed capacity 30

. . 47

52

. 76

.. 77

.. 77

. 78

. 79

. 79

80

80

. . 82

83

83

84

90

.. 101

.. 104

.. 105

.. 105

.. 109









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

OPTIMIZATION MODELS FOR CAPACITY PLANNING IN HEALTH CARE
DELIVERY

By

Chin-I Lin

May 2008

C'I I!r: Elif Akgahl
Major: Industrial and Systems Engineering

Health care capacity planning is the art and science of predicting the quantity of

resources required to deliver health care service at specified levels of cost and quality.

Because of variability in the arrival of patients and in the delivery of health care services,

successfully meeting the demand for health care services is a daunting task that requires

an understanding of the inherent trade-off between its cost and quality of service.

In our work, we model the general health care systems as queueing stations and

incorporate queueing theory into an optimization framework. The queueing modeling

approach captures the stochastic nature of arrivals and service times that is typical in

health care systems. The optimization framework determines the minimum cost capacity

required to achieve a target level of customer service. The inclusions of queueing equations

and discrete capacity options result the capacity planning models in non-linear integer

programming formulations.

We develop effective solution algorithms to obtain high quality solutions particularly

for realistic-sized problems. For the analysis of underlying queuing systems, we either

use available results from the literature or develop approximations. For the solution of

optimization models, we employ network optimization, meta-heuristic, and Lagrangian

relaxation approaches to develop effective solution algorithms. We present results from

extensive computational experiments to demonstrate the computational efficiency and

effectiveness of the proposed solution approaches.










CHAPTER 1
INTRODUCTION

Capacity planning decisions are important to any industry, especially to health care

industry because not only it relates to the management of highly specialized and costly

resources (i.e., nurses, doctors, and advanced medical equipment), but also it makes a

difference between life and death in critical conditions. Health care capacity planning

involves predicting the quantity and particular attributes of resources required to deliver

health care service at specified levels of cost and quality.

According to the resource availabilities, capacity planning can be classified into three

levels including strategic, tactical and operational levels. In the strategic level, decision

makers focus on the long term capacity decisions such as locations of medical facilities or

sizes of medical facilities and workforce. The tactical level capacity plan concerns policies

which could improve the service performance, on the perspective of either health care

providers or receivers, by re I11.~ n .111. expanding or downsizing the current resources.

In contrast, the capacity plan on the operational level concentrates on how to meet

demand by using the existing resources through appropriate methods such as scheduling or

overrun.

In our study, we focus on long-term level health care capacity planning decisions,

where the knowledge of the performance of the system at steady state is sufficient. In

addition, we model the health care service facilities as queueing systems to take the

stochastic patient arrivals and length of stay into account and measure the system

performance. The purpose of our study is to incorporate queueing models into optimization

framework to determine the optimal capacity level that minimizes cost while maintaining

a desired level of performance on patients' service quality and financial and/or operational

restrictions.

In our work, we study four different, practically relevant, long-term capacity planning

problems. C'!. Ilter 2 introduces the aggregate hospital bed ** i'-'r -;1 / rher':..t. (AHBCP)










problem, in which we model the hospital as a G/G/c queueing system with a single bed

type and a single patient class. Using the approximation from Bitran and Tirupati [16, 17]

to estimate the patients' expected waiting time, we determine the optimal bed capacity

plan over a finite planning horizon utilizing a network flow approach.

C'!s Ilter 3 presents the health care tersam -r'/'- I 1'1.r t. e.. t. (HC TCP) problem,

which finds applications in hospital emergency room and outpatient clinical settings.

The underlying queueing system is more complex than the one used for AHBCP. In

particular, we consider a system with two classes of patients with different priorities and

two types of health care teams with patient class-dependent service rates. Moreover, there

is o t;,l,,l,, I,.n, ;, ? -1.1 1.76.7.1 / between the teams, i.e., while one team can provide service

to both classes of patient, whereas the other team can serve only one class of patient. For

this queueing system, we develop an approximation approach to compute the average time

that a patient spends in the system for each patient class. Then, we integrate the results

from approximation method into an optimization model to make long-term health care

team capacity decisions.

C'!s Ilter 4 states the hospital bed allocation (HBA) problem, which is an extension the

AHBCP problem. After the .I negate bed capacity is specified, the next step involved is

concerned with the allocation of .l__-a egate bed capacity among different medical care units

(1!CI~s). We model each MCIT in a hospital as a Af/Af/c/c queueing system to estimate

the probability of rejection when there are c beds in the unit. We develop an optimization

model to allocate the .I negate bed capacity across different MCI~s.

O nI Ilpter 5 details the emer i. ,:..;; room services f r. .:1.:1 location and -r'/'- I 1'7-' : ': ':

(ERSFLCP) problem, in which each facility is modeled as a Af/Af/c/c queueing system.

We construct a facility location model, which simultaneously determines the number

of facilities opened and their respective locations as well as the capacity levels of the

facilities so that the probability that all servers in a facility are busy does not exceed a

pre-determined level.











Last, C'!s Ilter 6 provides a summary of the four problems we have investigated and

0---- -_ -r future research directions.









CHAPTER 2
AGGREGATE HOSPITAL BED CAPACITY PLANNING

2.1 Introduction

Capacity planning is central to the pursuit of balancing the quality of health care

delivered with the cost of providing that care. Such planning involves predicting the

quantity and particular attributes of resources required to deliver health care service at

specified levels of cost and quality. In general, successful health care capacity planning

must address a variety of issues, including the duration of the planning horizon (i.e.,

operational, tactical, and strategic), the level of care provided (i.e., primary, secondary,

and tertiary), the type of care (i.e., inpatient and/or outpatient), the amount, capability,

cost, and types of available or desired resources (i.e., doctors, nurses, technicians, medical

and clinical support staff, facilities including buildings, rooms, beds, parking spaces,

medical diagnostic and monitoring equipment, or any other element that constitutes

an "input" to the delivery of health care) as well as the customer service metrics or

performance measures expected for the facility (e.g., patient length of stay, likelihood

of full capacity where all inpatient beds or examining rooms are occupied, utilization of

providers and facilities, and financial performance such as having expenses within or below

budget).

While capacity planning has challenged health care decision makers and researchers

for decades [90, 91, 101], there is a renewed sense of urgency to address this problem.

In addition to the perennial struggle between the continually increasing costs of highly

specialized and scarce inputs (i.e., skilled and flexible staff, advanced clinical and medical

technology and equipment, physical space and supplies) and declining government

and private reimbursements [89, 96], the demand for inpatient care has been growing

substantially. According to the American Hospital Association (AHA), while average

length of stay (ALOS) remained unchanged at 5.7 doi~ all community hospital volume

statistics increased from 2002 to 2003: inpatient admissions by 0.C' to 34.8 million,









total hospital-based outpatient visits by 1."' to 563.2 million, emergency department

visits by 1.0' to 111.1 million, adjusted average daily census (i.e., average number of

inpatients and outpatients receiving care per d .;) by 0.C' to 894,000, and average

inpatient occupancy rate increased by 1.C,' to 66.>' [7]. However, the number of hospitals

of all types decreased by 30 to 5,764, there were 32 fewer community hospitals, and 8,000

fewer community hospital beds in 2003 [7].

In this paper, we focus on .l__ regate hospital bed capacity planning decisions. We

develop a model to simultaneously determine the timing and magnitude of changes in bed

capacity that minimizes capacity cost (including the cost of changing capacity as well as

the cost of operating capacity) while maintaining a desired level of facility performance

(e.g., limiting a patient's expected delay before being admitted to a bed and keeping

expenses within budget) over a finite planning horizon. We divide the planning horizon

into discrete time periods of equal length, and assume that the system achieves steady

state in each of these intervals. This allows us to use queuing methodology to analyze

system performance, but this typically leads to nonlinear equations in our formulation. As

hospital bed capacity must be integer valued, our planning model is a large-scale nonlinear

integer optimization model that minimizes total cost while achieving a targeted level of

system performance. We show that some practical considerations lead to simplifications in

the model, which leads to a network flow formulation for the problem that can be solved

in polynomial time.

A variety of problems that arise in the context of transportation, finance, ]rn Ilrirll. (ilirin

and service systems can be modeled as network flow models [2]. A network is a collection

of capacitatedd or uncapacitated) nodes and (directed/undirected and capacitated/uncapacitated)

arcs, where the arcs link one node to another and carry flow from one node to another.

Well-known network flow models are the shortest path, maximum flow, and minimum

total cost flow formulations, for which efficient solution algorithms exist [2]. In our work,

we show that the capacity planning model we consider can be transformed into a shortest










path model, where the objective is to find the path from the source node to the sink node

with the shortest length (i.e., the minimum cost bed capacity plan from the current period

to the final period of a given planning horizon).

The remainder of this paper is organized as follows. Section 2.2 provides a brief

overview of the history and current research in hospital bed planning. In Section 2.3, we

describe the system and give three models for planning hospital bed capacity. In Section

2.4, using data from a medium-sized medical center, we provide a computational study to

illustrate how the model formulations can he used and how changes in problem parameters

can affect our ability to obtain an optimal solution. Section 2.5, offers several practical

extensions of our model. Last, we give concluding remarks and discuss future research

directions in Section 2.6.

2.2 Literature Review

During the 1990s, many hospitals in the United States reported having too many

beds and were exploring strategies to reduce space [10, 15, 29, 45, 46, 50]. Less than a

decade later, in part due to renewed growth in demand for inpatient services [7, :31], most

hospitals are currently facing considerable space and resource restrictions forcing them

to contemplate expensive renovations and/or new construction projects to increase bed

capacity [11, :31, 57]. However, whether hospitals, in fact, need the additional capacity

appears to be unresolved [10, 45]. On one hand, increased inpatient admissions coupled

with fewer hospitals and fewer hospital beds would support the argument in favor of

capacity increases [7, :31]. Conversely, level or decreasing average length of stay and a

corresponding decrease in the average inpatient occupancy rate may imply that existing

capacity is sufficient [10]. Regardless, determining the optimal number and organization of

hospital beds continues to be a challenge.

The ability to anticipate bed demand and match it with the appropriate bed supply

is critical to effective bed planning. Health care decision makers know that both will

be influenced by a number of factors. Factors internal to the decision makers include









containing the costs associated with operating, contracting, and expanding current bed

capacity, reducing bed assignment waiting, maintaining quality of care when patients

are placed in inappropriate units (e.g., an intensive care patient may have to be placed

in a cardiac unit), eliminating emergency department bottlenecks (i.e., keeping patients

in the emergency department after initial treatment due to unavailability of beds in

the appropriate care unit), and reducing the probability of diverting patients to other

hospitals due to lack of bed capacity [45, 46]. Externally, factors facing decision-makers

include atypical changes in community health (e.g., severe flu strains), annual Is id .~l17-<

(e.g., Thanksgiving), and the availability, size, and composition of appropriate medical

personnel. Historically, starting with the Hill-Burton Act of 1946, bed capacity planning

has tended to be based on target occupancy levels (TOLs) that are assumed to reflect

capacity levels that achieve an appropriate balance of cost, patient d. 1 .1- and resource

utilization. TOLs are derived using analytic models of typical hospitals in different

categories and are based on acceptable patient d. 1 .1-< for different services. However,

Green and Nguyen [46] use queuing models to investigate the relationship between

occupancy levels and delay, and concluded that using TOLs as the primary determinant of

bed capacity is inadequate and may lead to excessive d. 1 .1-< for beds. In particular, a TOL

does not necessarily correspond to a desired service level, and there is a need to quantify

the desired service level and measure its cost implications accurately.

Ryan [95] provides a capacity expansion model with exponential demand and

continuous time intervals and continuous facility sizes. In the context of health care

pl1 ....lr however, it is more realistic to model capacity expansion as the product of

limited, discrete choices as routine planning sessions (e.g., bimonthly or quarterly) where

capacity increases or decreases occur in some fixed bed amount such as a 20-bed unit.

Bretthauer and Ci~ti [26] model a general health care delivery system as a network of

queuing stations and incorporate the queuing network into an optimization framework to

determine the optimal capacity levels subject to a specified level of system performance










(e.g., average total time spent at the facility). They use an algorithm combining

branch-and-hound with outer approximation cutting plane method to solve the nonlinear

optimization problem with discrete variables, but a disadvantage of this algorithm is that

in the worst case the algorithm could require complete enumeration of all integer solutions,

leading to very large solution times.

2.3 Problem Formulation

In the bed capacity planning problem, we start with a planning horizon of length

T indexed by t=1, 2, ..., T. Let A, denote the .I__regate patient arrival rate in period

t, 1/p- he the ALOS per patient, and the service rate per bed per dwi is given by ft or

1/ALOS and the service rate per bed over period t is ys,. In practice, there are alternative

patient streams (including admissions front the emergency department, admissions front

referrals, and elective admissions) for each of which the typical length of stay may be

different. As the objective of our work is to provide an .I__-aegate planning tool for bed

capacity nianagenient, we assume that the arrival rates for different patient streams can he

combined and a representative value for the average length of stay per patient (regardless

type of services required by the patient) can he determined. Note that while ALOS has

been relatively stable over time [7], the actual A, for a given facility will not he known

until the demand presents itself. Therefore, for the purposes of capacity planning, A,

can he forecasted by a seasonally adjusted trendline, for example [32]. Let nt denote the

nmaxiniun allowable expected delay for a patient before the patient is admitted to a bed in

period t. We note that the number of beds in the system in a given period can he limited

due to other resource limitations including as the physical size of the facility and/or the

amount and type of personnel available. Let co be the initial bed capacity in the hospital.

Last, there is a budget limit on the amount of monetary resources that can he allocated to

purchasing additional bed capacity denoted by yt.

We have three types of decision variables. Let .r, he number of beds in period t. Let

.r,+ he the amount of increase in bed capacity at the beginning of period t, and r,- the










amount of decrease in bed capacity at the beginning of period t. Let f (xt, Xt, I-t) denote

the expected patient waiting cost as a function of number of beds xt, patient arrival

rate Xt, and average service rate Pt in period t. Similarly, let g(xt_l, xt) denote the cost

of changing bed capacity from xt-1 to xt (i.e., the cost of increasing or decreasing the

existing bed capacity) in period t. Let h(xt) denote the cost of operating xt beds in period

t. Finally, the expected d+. i- for a patient in period t is a function of number of beds

xt, patient arrival rate Xt, and service rate per bed pt, denoted by w(xt, Xt, I-t). We can

formulate the ..-:-o negate hospital bed capacity planning (AHBCP) problem as a nonlinear

integer programming formulation as follows:

I~i ? )fZ!LT T T)
t= 1 t= 1 t= 1
subject to
w(xt, Xt, I-t)l < t Vt (2-2)
Xo = co (2-3)
xt-1 + x, xt- = xt Vt (2-4)
g(xt-1, xt)l xt, x,+, xt- are discrete varibles V t (2-6)

The objective function (2-1) minimizes the total cost of patient waiting, changing

the bed capacity, and operating the existing bed capacity. Constraint (2-2) imposes

a maximum allowable limit on the expected patient waiting. For example, in order to

quantify the expected delay for a patient to be admitted to a bed, we assume that the

hospital can be represented as a GI/G/s queueing system and use the expected waiting

time approximation provided by Bitran and Tirupati [16, 17] to calculate a patient's

expected wait for a hospital bed. Constraint (2-3) sets the initial bed capacity while

constraint (2-4) is a flow balance equation stating that the number of beds available in a

period is equal to the number of beds available in the previous period plus the increase in

bed capacity minus the decrease in bed capacity. Constraint (2-5) is the budget constraint










that limits the amount of the funds allocated to changing capacity. Last, constraint (2-6)

ensures that the number of beds available and changes in bed capacity are integer valued.

2.3.1 Restricted Bed Capacity Planning Problem

It should be readily apparent that the number of integer variables associated with the

AHBCP problem could be quite large as there is no restriction on how many beds can be

added or removed from service. For example, community hospitals may have 500 or more

beds [7]. In practice, bed capacity is increased or decreased in batches, and is typically

changed in integer multiples of a base value, or-, in multiples of 10 or 25 corresponding

to the size of a unit. As a result, there are only a limited number of choices for changing

capacity in each period. Therefore, constraints that capture the change in capacity can be

replaced by a set of discrete alternative constraints, requiring that only one alternative is

chosen in the solution for each period. Then, the original non-linear integer programming

problem becomes a nonlinear binary (i.e., zeroone) integer programming problem, which

we refer to as the restricted bed capacity planning (RBCP) problem.

In the RBCP problem, we are given a base value of B in multiples of which the bed

capacity can be increased or decreased and we let a be the number of possible distinct

levels of capacity increase or decrease. That is, given bed capacity c in period t, the bed

capacity in period t + 1 can be one of (c nB)+, (c (n 1)B)+, ..., (c B)+, c, c + B, ...

c +(n -1)B, c +nB, where (x)+ = max{0, x}. We assume that all acquired new additional

capacity is available and becomes effective capacity in the same period. Let zgt = 1 if the

available bed capacity is increased by iB at the beginning of period t for i=1, 2, ..., n; and

0 otherwise. Similarly, let zy = 1 if the bed capacity is decreased by iB at the beginning

of period t for i=1, 2, ..., n; and 0 otherwise. We can now formulate the RBCP problem as

a nonlinear zeroone integer programming problem as follows:

T T T
minl Cf (xt, At, Pt) + g(xt-1, ) + h(xt) (2 7)
t= 1 t= 1 t= 1
subject to









w(xt, Xt, I-t)l < t Vt (2-8)
Xo = co (2-9)

xt-1 + i~zg i :r t V~t (2-10)
i= 1 i= 1

zg a+ zy < 1 Vt (2-11)
i= 1 i= 1
g (xt-1, xt)l xt > 0 V t (213)
zg, g e 0, } V t(2-14)

As in the AHBCP problem, objective function (2-7) minimizes the total cost of

patient delay, changing the bed capacity, and operating the existing bed capacity,

constraint (2-8) imposes a maximum allowable limit on the expected patient delay,

constraint (2-9) sets the initial bed capacity, and constraint (2-10) is a flow balance

equation. Constraint (2-11) ensures that only one choice for changing the capacity is

allowed in each period. Constraint (2-12) imposes the budget constraint on the amount

of money allocated to changing bed capacity. Constraints (2-13) and (2-14) ensure the

nonnegativity of the bed capacity level and capacity level selection decision variables,

respectively.

An attractive feature of the RBCP problem is that a network representation can be

developed. Consider a T-partite graph with T lI-;-rs each representing a time period t=1,

2, ..., T in the planning horizon. Let (t, c) denote the system when there are c beds in

period t. Let C(c) be the set of reachable capacity levels in the next period if the capacity

in the current period is c, and we have C(c) = {(c nB)+, (c (n 1)B)+,..., (c -

B)+, c, c + B, ..., c + (n 1)B, c + nB}. Let St be the set of all capacity levels reachable

in period t from all capacity levels in period t 1. Let d, be a superficial source node

connected only to node (0, xo) with zero arc length. Node (0, xo) represents the beginning

state where there are xo beds in the hospital at time t=0. Let (0, xo) be connected to all

nodes (1,x') for x' E C(xo). If w(x', Az, pt) < atl (i.e., the patient waiting time constraint

is not violated) and g(xo, x') < yl (i.e., the budget constraint is not violated), then the










length of these arcs are given by f(x', Ai, pt) + g(xo, x') + h(x') (i.e., the expected patient

waiting cost with x' beds in the system, total cost of changing the bed capacity from xo to

x', and cost of operating x' beds). However, if either constraint is violated, then the length

of the corresponding arc is set to M~, where M~ is a very large number. Similarly, let each

node (t, x) for x E St and t=1, 2, ..., T-1 be connected to (t + 1, x') for x' E C(x) with

length f(x', Xt+l, I-i+i) + g(xt, x') + h(x') if w(x', Xt+l, I#<+l) < a~t+l and g(xt, x') < 7t+l,

and M otherwise. Last, let each node (T, x) for x E ST be connected to a superficial sink

node dt with an are of zero length.

Figure 2-1 provides an example of the network representation for the RBCP problem

where co=300, B=25, n=1, and T=4. In this figure, a path from the superficial source

node to the superficial sink node represents a plan for the bed capacity over the planning

horizon. The shortest path without containing any arc with cost M~ yields the capacity

plan with total minimum cost that obeys the patient waiting time and budget constraints

over the planning horizon. If no such path can be found (i.e., the shortest path contains at

least one arc with cost M~), then the problem is infeasible and no capacity plan that obeys

the waiting time and budget constraints over the planning horizon can be found.


Figure 2-1. Network flow representation for RBCP with co=300, B=25, n=1, and T=4









Recalling that there are n distinct levels to increase or decrease capacity, the general

network flow representation representation for the RBCP problem has 2nt + 1 nodes in

1.s.- rt for t=1, 2, ..., T. Therefore, there are a total of nT(T + 1) + T + 2 nodes (including

the superficial sink and source nodes) and the shortest path for the RBCP problem can be

found in O(n2 4) time using Dijsktra's algorithm [2].

2.3.2 Restricted Bed Capacity Planning Problem with Shuttering

In the RBCP problem, we assume that the cost of increasing or decreasing bed

capacity is uniform. In practice, however, decreasing bed capacity can be achieved by

shuttering existing bed capacity. That is, a hospital unit is closed and the personnel may

be reassigned to other units in the hospital or laid off, thereby, reducing the effective bed

capacity. On the other hand, increases in bed capacity can be accomplished two v- .1-<. If

the existing capacity is larger than the effective capacity implying that shuttered capacity

is available, then restoring a shuttered unit into operation by reallocating personnel to

this unit can increase bed capacity. However, if the existing capacity is equal to the

effective capacity implying that no shuttered capacity is available, then bed capacity

can only be increased through a capital investment to open a new unit and purchase

new beds. We can incorporate this practical concern into our formulation easily by

changing the definition of the objective function by keeping track of the effective and

existing bed capacity in the hospital. We now distinguish between two types of capacity

changes, where g(xolx', |2')> is the cost of changing effective capacity from xo to x' via

shuttering and the existing bed capacity from x to x' via acquiring additional capacity

where x' > max~x', x}. As before, we assume that all acquired new additional capacity

becomes effective capacity in the same period. The formulation is still a nonlinear zeroone

integer programming problem, which we refer to as the restricted bed capacity planning

with shuttering (RBCPwS) problem.

As with the RBCP problem, a network representation can be developed for the

RBCPwS problem. Consider a T-partite graph with T lIn-;-rs each representing a time









period t=1, 2, ..., T in the planning horizon. Let (t, clc) denote an effective capacity

of c and an existing capacity of c in time period t, and c < c. Let Cl(clc) denote the

set of reachable capacity levels via shuttering only, C2(~clc) ~lthe set of rea~chalev capaicit~y

levels by acquiring new additional capacity and C(clc) =Cl(clc) U C2(clc) lthe set~ of

all reachable capacity levels in the next period if the effective and existing bed capacity

in the current period are c and c, respectively. If we have c + nB < c, then we have

C, (clc) = {(((c nB)+I|a), ..., ((c B)+I|a), (cl c), (c + BI c), ..., (c + nB I )} and C2 (C C> = (8.

Also, if we have c < c < c+ nB, we have Cl(cl c) = {((c- nB)*|a)c,..., ((c B)*|a)c, (clc), (c+

Blc),..., (clc)} and C2(clc) =( {(+ BI ),..., (c +nB I )}. Again, let d, be a superficial source

node connected only to node (0, xo Ix) with zero arc length. Node (0, xo Ix) represents

the beginning state where there are x beds in the system and xo in operating condition

at t=0. Let (0, xo|2) be connected to all nodes (1, x'|2) in C(xolx). Provided both the

patient waiting time constraint and the budget constraint are not violated, then the

length of these arcs are given by f(x', Ai, pt) + g(xolx, x'|2') + h(x') (i.e., the expected

patient waiting cost with x' beds in the system, total cost of changing the effective bed

capacity from xo to x' via shuttering and the existing bed capacity from x to x' via new

bed acquisition, and cost of operating x' beds). If either of these constraints is violated,

then the length of the corresponding arc is set to M~. Similarly, each node (t, x|2) for

(x|2) E St and t=1, 2, ..., T-1 be connected to all nodes (t +1, x'|2') in C(x|2') with length

f(x', Xt+l, I-t+l)+g(x|2, x'|2')+h(x') if w(x', Xt+l, I#<+l) < a~t+l and g(xlx, x'|2x') < 7t+l, and

M~ otherwise. Finally, let each node (T, x|2) for (x|2) E St be connected to a superficial

sink node dt with length zero.

Figure 2-2 provides an example of the network representation for the RBCPwS

problem where co=275, co=300, B=25, n=1, and T=4. For ease of exposition, the thin

arcs represent opening, maintaining, or shuttering of existing capacity, whereas the thick

arcs represent the acquisition of new capacity. In this network, a path from the superficial

source node to the superficial sink node represents a plan for the bed capacity throughout










the planning horizon. As before, the shortest path without containing any are with

cost Af in the network yields the capacity plan with total nxininiun cost that obeys the

patient waiting time and budget constraints throughout the planning horizon by allowing

capacity changes via shuttering and/or acquiring additional capacity. If no such path can

he found (i.e., the shortest path contains at least one are with cost Af), the problem is

infeasible and no capacity plan that obeys the waiting time and budget constraints over

the planning horizon can he found.


Figure 2-2. Network flow representation for RBCPwS with co=275, B=25, n=1, and T=4

Recalling the RBCP problem, we have specified the number of arcs and nodes in

the network to determine the time to obtain the optimal solution. However, for the

RBCPwS problem, since existing capacity can he increased further through capital

acquisition, the analysis becomes slightly more complicated and dependent on the initial

state (i.e., the amount of effective and existing bed capacity). If no additional capacity










has to be purchased throughout the planning horizon, then the RBCPwS and RBCP

networks are identical and the size of the RBCP network is a lower bound on the size

of the RBCPwS network. If additional capacity has to be purchased in a period, at

most (t 1)n2 nodes can be added to the network in that period. Hence, if the initial

effective capacity is equal to the existing capacity, then there can be at most a total of

nT(T + 1) + T + 2 + T(T + 1)(T 1)n2/6 nodes (including the superficial sink and source

nodes) in the network, and the shortest path can be found again in O(n4 6) time using

Dijsktra's algorithm [2].

2.4 Illustration of the Model

In this section, we illustrate the practical applicability and computational behavior of

our model through two experiments. In the first experiment, we illustrate how our model

can be used to develop bed capacity plans. In the second experiment, we quantify the time

(in CPU seconds) needed to obtain optimal solutions. In both experiments, we use the

RBCPwS formulation and its associated network model.

2.4.1 A Representative Decision-Making Scenario

To set the stage for the computational experiments that follow, we present a

representative decision-making scenario based upon a real-world application of our model

to a medium-sized, non-government, not-for-profit, general medical and surgical medical

center. Administration at this facility provided us with information about their facility,

capacity planning decision-making processes, and facili T- specific data for bed size, bed

operating cost, bed acquisition cost, and quarterly patient demand. However, note that at

the request of the facility's administration, the data presented here have been modified to

protect their identity, but are representative of similar-sized facilities.

This facility would like to determine an optimal bed capacity plan for the next eight

quarters, corresponding to its operational, budgetary, and strategic planning periods.

Because capacity planning may involve a substantial capital commitment, it is imperative

that any capacity expansion plan be carefully developed and justified based upon the










facility's current and expected demand. The facility's decision makers would like to

minimize the total capacity cost associated with the cost of changing capacity as well

as the cost of operating capacity while ensuring that the average time a patient should

wait for a bed does not exceed one hour (an internal benchmark for bed assignment).

At this facility, both existing and effective bed capacities are :350 heds, capacity change

can occur in increments of 10 heds, and there are two levels of capacity increase (i.e.,

initially, bed capacity can range from :330 to :370 heds, in 10 hed increments). Based

on information from the facility's administration, it costs $2,000/d is- to operate an

effective bed, $2,500/d is- to either shutter an effective bed or reactivate a shuttered bed,

and $200,000/hed to expand bed capacity through capital investment. Last, because

of seasonal migration (or -a! 0.-- hirds"), demand at the facility can be highly variable

throughout the year, and we were provided with data and guidance on values related to

patient arrival rates and service times.

2.4.2 Experiment 1 An Application of the Model

The intent of this experiment is to illustrate how our network flow model can he used

to make bed capacity decisions and generate a T-period capacity plan. Our base scenario

was described in Section 2.4.1, and we refer to it as S1. Table 2-1 lists the relevant

parameter settings for S1, and other experimental scenarios relative to this scenario are

given in Table 2-2.

At the outset, we provide an estimated range of demand for the facility over the

planning horizon. Normally, a single seasonally adjusted trendline would be computed

to forecast the patient arrival rate based on historic demand data. Instead, to illustrate

the extent of variation in demand, Figure 2-3 di pF.T-4~ a set of simulated patient arrival

rates over the planning horizon based upon the scenarios given in Table 2-2. We note

that some of the parameter changes directly impact the patient arrival rate, and different

patient arrival rates are generated. In S1, S:3, S4, S5, and S6, the changed parameters do

not impact the arrival rate function, so these scenarios have identical arrival rates. (Note










Table 2-1. Parameter settings for the base scenario, S1
Parameter Value
Length of the planning horizon T = 8 quarters, t= 1, 2, ..., 8
Forecasted demand per time period t Xt = Smod(t,4)U(a + bt) where as is a
quarterly seasonal index (i.e., sl=0.8,
s2=1.0, s3=1.2, and s4=1.0), a is a
uniformly distributed random number
(i.e., u~- U[0.8, 1.2]), a=6,400, and b=128
Initial existing bed capacity co = 350
Initial effective bed capacity c = 350
Number of levels of capacity increase or n = 2


decrease
Incremental amount of capacity change
Cost to operate an effective bed
Cost to shutter an effective bed
Cost to reactivate a shuttered bed
Cost to acquire a new bed
(i.e., expand capacity through capital
investment )
Coefficient of variation for arrivals
Coefficient of variation for service
Maximum expected delay per patient
Cost of waiting
Service rate


B =10
$2,000/bed
$2,500/bed
$2,500/bed
$200,000/bed


cat = 0.5
est = 0.5
at = 1 hour
$300/hour
I-t = 15.8 patients per bed


Table 2-2.
Scenario
SO
S1
S2
S3
S4
S5
S6


Scenario descriptions for experiment 1
Description
Level demand
Base scenario
Increased rate of demand
Higher demand variability
Higher service variability
Higher cost of waiting per patient
Smaller maximum expected del li per patient


Parameter change
b =

b = 256
cat = 2.0
est = 2.0
$1,200/hour
a~t = 0.25 of an hour


that with S3, higher variability in the arrival rate impacts the performance constraint for

average waiting time, not the arrival rate function.) For SO and S2, the patient arrival rate

function has no trend and a higher trend compared to S1, respectively. Hence, arrival rates

generated for these scenarios are significantly different from each other and S1.

We have implemented our network flow approach using the C++ programming

language and solved for the scenarios using a personal computer with 3.0 GHz Pentium IV





-* SO 350 330 310 290 270 250 230 250 250
-CS1 350 330 310 320 300 280 290 300 310
-A-S2 350 330 310 320 300 310 330 350 370
S3 350 330 310 320 300 280 300 310 320
--K-S4 350 330 310 320 300 280 300 300 320
--+-- S5 350 330 310 330 310 290 300 310 320
------S6 350 330 310 320 300 280 300 300 310

Time periods


Figure 2-4. Optimal capacity plans for experiment 1


11000




j 9000--

18 8000--

~i7000--

.k~ 6000-- -

S5000--
$-*- SO
4000 -- I S1,S3,S4, S5, S6
--A- S2
3000I IIIIII
12 3 4 5 6 7 8
Time periods


Figure 2-3. Patient arrival rate for experiment 1


processor and 512 MB R AM memory. We obtained the optimal solution for each scenario

and the results are depicted in Figure 2-4, where each line represents the optimal capacity


plan that corresponds to one of the seven scenarios.


375

35)

S325





275
-0


SS()

S1

-A- S2

- M-- 3

--rc- S4

--+-- S5

------ S6










In considering Figure 2-4, we have the following observations. For S1, we first observe

a general reduction in the bed capacity, then a gradual increase near the end of the

planning horizon. The initial bed capacity seems to be higher than needed, and as a

result, the bed capacity is reduced to reduce total costs over the planning horizon while

maintaining the average waiting time constraint. Of course, when the demand increases

due to the underlying trend, the bed capacity is increased. When demand is level as in

SO, a lower envelope is formed relative to the base case (i.e., the bed capacity for SO is less

than or equal to the base case). Similarly, with an increased rate of demand as in S2, an

upper envelope is formed relative to the base case. With increased variation as in S3 and

S4, the optimal capacity plans are similar to S1's capacity plan but tend to require higher

capacity when the arrival rate is increasing. When the arrival rate increases in periods

6, 7, and 8, because the higher arrival variability and higher service variability affect the

average waiting time constraint, more capacity is required to keep from violating this

performance constraint. Likewise, with a higher cost of waiting per patient as in S5 or a

tighter average waiting time performance constraint as in S6, the optimal capacity plans

tend to require capacity slightly higher than the base case. Not surprisingly, the net result

of this experiment indicates that optimal bed plans are driven substantially by changes in

demand. While health care decision makers may not he able to affect overall demand for

their services, if they can reduce variability in arrivals [78] or are willing to tolerate a less

stringent performance constraint, less capacity will be required.

2.4.3 Experiment 2 Assessing the Impact of Problem Parameters

As we have discussed earlier, an upper bound on the size of the network (i.e., number

of nodes in the network) representing a problem instance of RBCPwS can he characterized

in terms of the number of levels for capacity increase or decrease and the number of time

periods in the planning horizon. The ratio of the effective bed capacity to existing bed

capacity impacts the size of the network. The size of the network can also be used to

quantify the computing time required to obtain the optimal solution. The time required










to build the network and find the optimal solution may change as the number of levels

increases, the planning horizon length increases, or the ratio of effective to existing bed

capacity changes. In order to illustrate the change in computational time, this experiment

has two parts: 1) the impact of effective to existing bed capacity and 2) the impact of

changes to the number of levels of bed capacity and the length of the planning horizon.

In the first part of this experiment, we fix the number of levels to vary bed capacity

and the duration of the planning horizon in addition to some other problem parameters

constant and examine the impact of different ratios of existing to effective bed capacity.

Using the assumptions for the base case scenario, S1, from the previous experiment, we

consider ten different levels of the effective bed capacity in the interval [260, 350]. We

generated 30 random test instances for each of these levels and the summary results are

provided in Figure 2-5 and Table 2-3.

600

500-

8 400-

S300-

S200-

S100-


250 260 270 280 290 300 310 320 330 340 350 360
Initial number of effective beds

Figure 2-5. Number of nodes in the network as a function of initial effective bed capacity

In Figure 2-5 we depict the number of nodes in the network, and in Table 2-3 we

report the time to build the network and time to obtain the solution for each level of the

initial effective bed capacity. The number of nodes increases as the ratio of effective bed

capacity to existing bed capacity approaches one, and this behavior is clearly depicted

in Figure 2-5. However, in Table 2-3, we see that an increase in the size of the network










increases the time to build the network only slightly, and its impact on the time to obtain

the solution is almost negligible. Therefore, our solution method is robust to changes in

the problem size that are induced by the initial effective bed capacity.

Table 2-3. Summary statistics for the RBCPwS problem's solution time (in CPU seconds)


as
Initial level
of effective
bed capacity
260
270
280
290
300
310
320
330
340
350


a function of initial effective capacity
Time (in CPU seconds) to
Build the netowrk Obtain the solution
Min. Avgf. la~x. Min. Avgf. la~x.
0.0 0.0 0.2 0.0 0.0 0.0
0.0 0.0 0.1 0.0 0.0 0.0
0.0 0.0 0.1 0.0 0.0 0.0
0.0 0.0 0.1 0.0 0.0 0.0
0.0 0.0 0.1 0.0 0.0 0.0
0.0 0.0 0.1 0.0 0.0 0.0
0.0 0.0 0.1 0.0 0.0 0.0
0.0 0.1 0.1 0.0 0.0 0.0
0.1 0.1 0.1 0.0 0.0 0.0
0.1 0.1 0.1 0.0 0.0 0.0


Total time
(in CPU seconds)
Min. Avgf. Alax.
0.0 0.0 0.2
0.0 0.0 0.1
0.0 0.0 0.1
0.0 0.0 0.1
0.0 0.0 0.1
0.0 0.0 0.1
0.0 0.0 0.1
0.0 0.1 0.1
0.1 0.1 0.1
0.1 0.1 0.1


For the second part of this experiment, we vary the number of levels to change bed

capacity as well as the duration of the planning horizon. We consider four different levels

to vary the bed capacity (i.e., n=2, 3, 4, 5) where capacity is increased in increments of

B=10, and three different time horizons (i.e., T=8, 12, 16) that correspond to two-, three-,

and four-year planning horizons. Therefore, we have 12 settings in total. For each setting,

we generated 30 random instances and for each of the instances, we also generated the

effective bed capacity as a fraction of the existing bed capacity. The summary results for

this experiment are provided in Table 2-4.

Fr-om Table 2-4, as the number of levels of capacity change and the length of the

planning horizon increases, the number of nodes in the network increases. The increase

in the number of nodes impacts the total time required to obtain the optimal solution.

However, a closer examination of the results reveals the increase in the number of nodes in

the network has a direct impact on the time required to build the network, and has almost

no impact on the time to obtain the solution. Only in the setting with the largest test

instances (i.e., n=5 and T=16) do we observe an increase in the time to obtain the






















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optimal solution. Even in that case, the nmaxiniun solution time is still less than a few

seconds. Therefore, our solution method is robust to changes in the problem size induced

by the number of levels of capacity change and the duration of the planning horizon.

2.5 Extensions

In this section, we discuss several extensions to our model. These extensions may

arise out of practical considerations associated with how our model addresses facility

performance .

In our model, we treat the performance constraints as a hard constraint. That is,

if a particular capacity level violates the performance constraint, then a solution with

that particular capacity level is not feasible, and is dropped front further consideration.

However, the performance constraint can he modeled as a soft constraint where we can

deliberately allow the violation of the performance constraint while incurring a penalty

cost to be added to the objective function. We can justify this constraint by noting that

lags typically exist between capacity levels so there might he periods of time where the

facility is operating above its typical utilization and the capacity expansion cannot occur

quickly enough to allow the organization to react to the change in demand. To illustrate

how our model can he reformulated with the soft constraint, let i, b e the amount of the

violation, s, he the amount of slack in the performance constraint, and xr(l ,) be thepenalty

cost incurred for violating the performance constraint in period t. Then, considering the

RBCP problem, objective function (2-7) would be replaced with:

T T T T
nxin f (.rXI,LI) A, p) g(. t1I .) + Ch(.r,) + x (r ) (215)
t= 1 t= 1 t= 1 t= 1

Similarly, constraints (2-8) and (2-13) would be replaced with:

w(.r,, A,, pt) ', + S, = nt Vt (2-16)
', = nmax {w(.rt, A,, p,) c0,, 0} V t (2-17)
s, = nmax {0, w(.r,, A,, p,), 0} V t (2-18)
.r,, '<,S 8 > 0 V t (219)










It is easy to observe that this modified version of the RBCP problem can still

be formulated and solved as a network flow problem. The variables i, and .st are

calculated by constraints (2-17) and (2-18), respectively, once w(.r,, A,, pi) is known.

The only modification of the network is to include the cost associated with violating the

performance constraint.

When evaluating a hospital, we recognize that the average waiting time to be assigned

a bed or having expenses within budget are not the only metrics to assess facility

performance. Indeed, it may be necessary to include measures for facility utilization,

likelihood of patient diversion, and the like. Regardless, we note that more performance

constraints can easily be added to the formulations for the BCP, RBCP, and RBCPwS

problems. An increase in the number of performance constraints does not increase the

time to obtain the solution significantly, as there is only a need to take these additional

constraints into account in setting up the network and assigning a large are cost in

case any of the constraints are violated. Therefore, our modeling approach is robust

and additional constraints can he considered without increasing the complexity of the

formulation significantly.

2.6 Concluding Remarks and Future Research Directions

We have presented a network flow approach to optimize bed capacity planning

decisions for hospitals. Our model incorporates the reasonable concerns associated

with determining hospital bed size, such as a finite planning horizon, an upper bound

on the average waiting time before a patient is admitted to a hospital bed, and a

budget constraint that limits the amount of money that can he allocated to changing

bed capacity. Further, our model accommodates capacity change through Ihot 1 fling

as well as expansion of bed capacity through new capital investment. Our series of

computational experiments illustrated both the ease of implementation of our model and

the sensitivity of the computational time required to obtain the optimal solution to several










problem parameters. We have also discussed extensions of our model in the form of soft

performance constraints and multiple performance constraints.

Our model is based on a generic view of a hospital where we have assumed that

the demand (i.e., patient arrivals) and service (i.e., beds) components are homogeneous.

From an .I__-oegate planning perspective, such uniformity may be acceptable. However, in

order to apply this research to operational decision support for health care delivery, there

are additional avenues of research worth pursuing. First, if cost depends on all previous

stages, for example, the cost of maintaining the beds depends not only on the number

of beds but also the duration of the beds are in the system, then the number of vertices

in the network will be exponential with respect to T and the optimal solution to the

network will not be solved with a polynomial time algorithm. Consequently, alternative

model formulations and solution techniques to determine the optimal bed plan would

be necessary. Second, recognizing that hospital beds are not identical, facility capacity

could be separated to distinguish the various specialties, with specialti- specific demand

rates, lengths of stay, and costs. In determining the average waiting time associated

with being assigned a bed, we have used closed-form approximations to calculate this

statistic. Therefore, we are implicitly assuming that this general distribution accounts

for different types of patients that require different types of hospital-based health care.

This may not necessarily be the case, and should be investigated further. Third, our work

can be expanded to include multiple types of patients (e.g., electives, admissions coming

through the emergency department, and referrals from physicians). Also, in estimating

the cost of patient waiting, we assume that this cost is identical regardless of patient

type. Clearly, for example, there should be different waiting costs associated placing an

emergency department admission in an appropriate unit versus an inappropriate unit.

As such, representations of patient waiting cost need to be developed in the presence

of congested, heterogeneous resources. Fourth, the current form of our model does not

account for the potential time delay that may exist between the decision to expand










capacity and actually starting to use the new capacity. Our current model formulations

would have to be amended to include the length of delay relative to a capacity expansion

(e.g., if a capacity expansion requires k time periods and we need to use the capacity in

period t, the decision to expand should occur on or before period t k) and reconciliation

of multiple capacity expansions over the planning horizon (e.g., if a capacity expansion

decision is made in period t, can the facility make another decision in subsequent periods

until t + k when the earlier decision contes into effect). However, because these capacity

expansion considerations would destroy the underlying polynonlially bounded network

structure of the current model, other solution methodologies would have to be developed.

Last, as evidenced by the current nurse shortage [3] and the ongoing debate regarding

nurse-to-patient ratios [4], the ability to use physical capacity hinges upon the availability

of suitable medical personnel. A natural extension of our model would be to incorporate

workforce planning to simultaneously determine the quantity and composition of the

health care resources to construct a comprehensive capacity plan.









CHAPTER 3
HEALTH CARE TEAM CAPACITY PLANNING,

3.1 Introduction

With a size of $1.9 trillion and growth rate of 7.9 percent [102], the health care

industry accounts for the largest sector of the economy in the United States (US). Despite

advances in medical technology and, thereby, the increasing use of medical diagnostic,

monitoring, and treatment equipment, the health care industry is highly labor-intensive.

According to the US Department of Labor, the health care industry provided 13.5 million

jobs in 2004, out of which 13.1 million jobs are for wage and salary workers and about

411,000 are for the self-emploi-v 1 [107]. It follows that personnel wages and salaries

account for the largest portion of the total expenditures for any health care facility. For

instance, hospitals spend on average about 54 percent of all expenditures on wages and

salaries [86]. Hence, health care personnel pI l ..... i.e., determining the appropriate mix

of health care personnel, needed to provide safe, effective, timely, and cost-efficient service

to patients [55], is an important problem.

In practice, both in inpatient and outpatient facilities, a health care team, comprised

of a group of health care personnel with different, and complementary, skill sets, provides

health care services to individual patients [34]. The members of the team (e.g. physicians,

physician's assistants, and registered nurses) are responsible to perform a set of tasks

required for the diagnosis, monitoring, and treatment of the patients. Some additional

tasks may have to be performed by other personnel (e.g., laboratory technicians,

radiological technicians, and radiologists) who are not a part of the health care team

but provide assistance for diagnosis and treatment. In the delivery of services by health

care teams, the safety and effectiveness of the service is ensured by the appropriate

selection of the service *'istl~.:.71;i of the team, whereas the timeliness and cost-effectiveness

of the service is ensured by the appropriate selection of the service *r'ist..Uti of the team.

The service capability of a team is characterized by the collection of the skills possessed










by each of the individual members of the team, whereas the service capacity of the team is

given by the total number of members included in the team.

In a health care facility, there are typically multiple types of health care teams each

with different capabilities. The patients that arrive to the facility are classified according

to their conditions (i.e., acuity levels) or medical requests. Based on this classification,

e ach p at ient i s as si gne d t o a health h care t eam and admit t ed t o an examinat ion /tre atment

(E/T) room that has the equipment necessary to provide the service needed by the

patient. As the set of tasks are shared among the members of the team and a typical

facility has multiple E/T rooms, a health care team usually serves multiple patients

simultaneously. For instance, while the team waits for the test results from the lab for

a patient, a registered nurse may be collecting specimens from another, a physician's

assistant may be suturing a wound of another, and a physician accompanied by a

registered nurse may be discussing a treatment plan with another. In our work, we

consider two particular settings where health care services are provided by teams.

Shands at Alachua General Hospital in Gainesville, Florida is a community hospital

that provides emergency medicine services. triage nurse, who determines the acuity level

of the patient and identifies whether the patient requires immediate (i.e., emergency) or

d.l I we 4 (i.e., urgent) care. The emergency care services are delivered by an emergency

care (EC) team, which is composed of physicians and registered nurses, and the urgent

care services are delivered by an urgent care (UC) team, which is also composed of

physicians and registered nurses. We note that the triage nurse does not belong to either

of these teams, but acts as a gatekeeper to route an arriving patient to either of the teams.

There are eight and two E/T rooms dedicated to the EC and UC teams, respectively.

Moreover, the EC team has the capability to attend to urgent care patients, but the

UJC team does not have the capability to attend to emergency care patients. Finally,

emergency care patients have pre-emptive Jp..-r,:70l over urgent care patients. That is, if

an emergency care patient arrives to the ED, while all E/T rooms dedicated to the EC










team are occupied by other patients and one of them is an urgent care patient, then

the emergency care patient preempts the urgent care patient out of the room and the

emergency care patient is immediately admitted to the room.

The Women's Clinic at the Student Health Care Center at the University of Florida

in Gainesville, Florida provides women's health care services. The outpatient clinic

(OC) serves not only non-acute patients, i.e., those who need routine services, but treats

acute patients also. The services for the diagnosis and treatment of acute illnesses and

abnormalities are provided by a physician (P) team, which is composed of a physician and

a physician's assistant. The routine clinical services are delivered by a nurse practitioner

(NP) team, which is composed of a nurse practitioner and a registered nurse. There are

four E/T rooms dedicated to the NP team and two rooms to the P team. Moreover, the

P team has the capability to attend to non-acute patients but the NP team does not have

the capability to attend to acute patients. Finally, acute patients have non-preemp~tive

Jp..- r,:70 over the non-acute patients. That is, if an acute patient arrives to the OC, while

all E/T rooms dedicated to the P team are occupied by other patients and one of them is

a non-acute patient, then the P team does not interrupt service to the non-acute patient

and the acute patient has to wait until an E/T room dedicated to the P team becomes

available.

In the settings we discussed above, the service capabilities of the health care teams

are fixed as dictated by the service needs of the patients, and personnel planning is mainly

concerned with determining the service capacity of the teams. In determining the service

capacity, however, administrators must take several additional facility capacity, budgetary,

and legislative constraints into account that limit the minimum and maximum total

number of each personnel type emploi-. I For instance, budget constraints may limit the

total number of physicians emploi-- II whereas legislated nurse-to-patient staffing ratios

may prescrobe a lower limit on the total number of registered nurses. Therefore, given the

lower and upper bounds on the total number of personnel with different skills that can be










emploi- II there is a finite number of team configurations that can he utilized by a health

care facility. For example, consider a health care team for which physicians and registered

nurses are required. Suppose that due to a budget constraint, the health care facility can

employ at most two physicians and four registered nurses at a time. Also, suppose that

due to a legislative constraint, the facility should employ at least one physician and two

registered nurses. Then, for this team, there are at most six feasible configurations, i.e.,

{ (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4) }, that administrators can choose from. Therefore, in

such settings, personnel planning is concerned with choosing an appropriate configuration

for each type of health care team.

In this paper, motivated by the practical settings we discussed above, we address

the long-term health care team service capacity planning (HCTSCP) problem in the

context of health care facilities where there are two patient classes and two types of teams.

We assume that the capability of each type of team is known, and the number of E/T

rooms allocated to each type of team is fixed over the planning horizon. Therefore, the

service capacity of a team can only be changed by modifying the configuration of the

team. 1\oreover, the service capacity of a team can he quantified by the service rate of

the E/T rooms dedicated to the team, which is the reciprocal of the average time that the

patients spend in the E/T rooms prior to discharge or transfer to another department.

We formulate a non-linear binary integer programming model to determine the service

capacity plan for the health care teams such that health care services are delivered in a

timely (i.e., the average time a patient from a particular class spends in the system does

not exceed a pre-specified threshold) and cost-effective (i.e., the total costs associated with

changing service capacity by hiring additional personnel, reassigning existing personnel

or laying off existing personnel and operating the service capacity are minimized) manner

over a given planning horizon considering some additional constraints.

To estimate the average time a patient from a particular class spends in the system,

we develop queuing models and decomposition based approximation (DBA) methods.










We note that in the practical settings we consider the steady state results obtained from

queuing analysis can be used in capacity pI ll onir_ as both systems are highly utilized,

and, hence, the systems reach to steady state quickly in both of the settings. To model

these settings, we consider queueing systems where there are two classes of patients and

two types of teams. Each team has a set of of E/T rooms dedicated to it, and there is

.I-i-inin.! it ic substitutability between the teams and their dedicated E/T rooms. That is,

class 1 patients (i.e., emergency care patients in the ED and acute patients in the OC) can

only be served in the E/T rooms dedicated to type 1 team (i.e., EC team in the ED and

P team in the OC), class 2 patients (i.e., urgent care patients in the ED and non-acute

patients in the OC) can be served in the E/T rooms dedicated to either type 1 or type

2 team. An arriving class 2 patient is admitted to a vacant type 1 E/T room only when

no class 1 patient waits in the system and no type 2 E/T room is available. In addition,

service rates are patient-class and team-type dependent, i.e., the time required to serve

a patient depends on both the class of the patient being served and the type of the team

that delivers the service. We consider both the preemptive and non-preemptive cases. Our

computational results illustrate that the DBA method and the capacity planning model

can effectively be used to make long-term personnel planning decisions.

The remainder of this paper is organized as follows. In Section 3.2, we review the

related literature. Section 3.3 presents our capacity planning model for HCTSCP and

show how it can be interpreted as a network flow model. We develop queueing models for

the ED and OC settings and present DBA methods to analyze these models in Section

3.4. In Section 3.5, we present results from our computational study that evaluates the

accuracy of the approximations from Section 3.4 and investigate the computational

performance of the formulation presented in Section 3.3. Section 3.6 includes a discussion

of the results and so__~--- -is future research directions.









3.2 Literature Review

Health care personnel capacity planning has been been studied for decades to address

long-ternt budgeting, niedium-ternt -r 11lo- and short-ternt scheduling decisions. A critical

examination of the analytical literature to date illustrates that a considerable amount of

effort has been allocated to the short-ternt personnel planning decisions. In particular,

the nurse rostering (also known as the nurse scheduling or the nurse staffing) problem

has been studied extensively, as nurse staffing costs account for a significant portion of

personnel costs in health care system [61]. Burke et al. [28] provide an excellent review of

the existing work in this area.

Analytical work that focuses on niedium- to long-ternt personnel planning is relatively

limited in scope. Schneider and K~ilpatrick [97] develop optimization models for personnel

planning in health care facilities. K~ao and Tung [62] present a linear progranining model

for the .I negate (nursing) workforce planning problem, which is later extended by Brusco

and Showalter [27] to account for the exogenous impact of nursing shortage. K~ropp and

Carlson [72] propose the integrated use of optimization and simulation modeling. Existing

work that uses simulation modeling is reviewed in Jun et al. [58]. Although the earlier

work in this area primarily focuses on the perspective of the health care provider by

placing an emphasis on nxinintizing the cost of personnel resources or nmaxintizing the

utilization of personnel resources, there is a need to take the patient's perspective into

account by considering the timeliness (e.g., nxinintizing of the waiting time and/or time

in system) or the availability (e.g., nxinintizing the probability of finding all servers busy

and being diverted to another service provider). To this end, we focus on nxinintizing the

sunt of capacity costs and cost of not providing service in a timely manner, while ensuring

that the average time in system for a patient does not exceed a pre-specified threshold.

To this end, we divide the planning horizon into discrete time periods of equal length and

assume that the system achieves steady state in each of these intervals. This allows us to

use queueing analysis to capture the stochastic behavior of the system and compute the










average time in system for the patients. Using the results of this analysis, we formulate

the HCTSCP problem as a mathematical programming model as in [5].

In the literature, there is a number of studies that investigate queueing systems

that are closely related to the setting we consider. Stanford and Grassmann [105] derive

the expected waiting time in a call center with unilingfual and bilingual servers serving

ill ll .s Ry- and nrlrairs..vl I--1ex-uage-use customers. A nrrairs..vi( i.-1 .vs- uage-use customer can

only be served by a bilingual server, and the type of a customer is not be known prior to

the first service. The service rates for all server types are the same, i.e., independent of

the customer type. Green [44] and Shumsky [99] also investigate expected waiting time of

a system with two types of customers and limited- and general-use servers. Both of them

consider the case where the service rates depend on the type of the server. Hence, the

distinguishing characteristic of the system we investigate is the dependence of the service

time on customer class for one of the server types. This seemingly a simple attribute of

the system leads to significant modeling challenges.

In queueing theory, to obtain the stationary probabilities of a system, the 1 its lin

approaches used include the matrix analytic methods [44, 60, 74, 75, 87, 105], power-series

algorithm (PSA) approach [19, 20, 54, 71], and DBA method [99]. The matrix analytic

methods formulate the system as a Markov chain to which the stationary probability xr

has the matrix-geometric form ar = xroR", where rate matrix R can be obtained through

an iterative algorithm [87]. The PSA approach first represents the stationary probabilities

of a system as power-series expansions of the traffic intensity of the system, and then

recursively solves for the coefficients of the power-series expansions by using the set of

stationary equations. K~ao and Wilson [64] compare the performance of the PSA and

matrix analytic methods with three iterative algorithms proposed in [60, 74, 75], and

conclude that the PSA performs extremely well in terms of computational speed, though

it may encounter difficulties in parameter settings which may lead to losses in accuracy.

Shumsky [99] proposes the DBA method. The DBA method first divides the state space










of the system into several regions, and then estimates the stationary probabilities in each

region by using simple approximate queueing models. Shumsky [99] illustrates that this

method generates performance measures rapidly with sufficient accuracy that can be used

in call center capacity decisions.In our queueing analysis, we also use this DBA method.

3.3 Problem Formulation

In the HCTSCP problem, we are given a planning horizon of length T, indexed by

t = 1, .. ., T. In each planning period, there is a limit on the funds that can be allocated

to changing service capacity of the teams in the facility, denoted by 7t for t = 1,..., T.

There are two patient classes, indexed by i = 1, 2. For each patient class there is an

upper limit on the number of patients in the facility, denoted by bi, and let b be the

two-dimensional vector representing these limits. In addition, there is an upper bound

on the average amount of time that a class i patient spends in the facility, denoted by asi,

for i = 1, 2. The forecasted arrival rate of class i patients in period t is denoted by Xie

for i = 1, 2 and t = 1,..., T, and let Xt be a two-dimensional vector associated with the

arrival rates of patients in both classes in period t.

There are two types of health care teams, indexed by j = 1, 2, and two types of

E/T rooms. Team type 1 can provide service to both patient classes with different service

rates in its dedicated E/T rooms, type 1, while team type 2 can only provide service to

patient class 2 in its dedicated E/T rooms, type 2. Let rj denote the number of E/T

rooms allocated to team type j for j = 1, 2, and r be the associated two-dimensional

vector representing the room allocation. As we discussed earlier, the service capacity

of each type of team can only be changed by modifying the configuration of the team

and can be quantified by the service rates of the associated E/T rooms. Given the lower

and upper bounds on the number of personnel with different skills that can be emploi-x &

in the facility and the types of personnel that must be included in a particular type of

team, let Ky denote the total number of possible configurations for team type j, indexed

by k = 1,...,Ky. Suppose that it is possible for the administrators to estimate the









service rate 8ijk, of type j E/T rooms when the associated team, i.e., team type j, has

configuration k, and the patient in the room belongs to class i for i = 1, 2, j = 1, 2 and

k = 1, .. ., Ky. We note that 012k = 0 for k = 1, K2 Since the class 1 patients cannot be

treated by the type 2 team or in the type 2 E/T rooms.

We have two sets of decision variables. Let xjkt take the value of one if for team type

j configuration k is selected in period t for j = 1,2, k = 1,...,Ky and t = 1,...,T,

and zero otherwise. Also, let xjt be a Ky-dimensional vector associated with the team

configuration decision for team type j in period t. Finally, let '., denote the service rate

for class i patients who are treated in type j E/T rooms in period t for i = 1, 2, j = 1, 2

and t = 1,..., T, and 4t be a 2 x 2 matrix associated with the service rates of the E/T

rooms in period t.

We let I, (#t, Xt, b, r) and as(#t, Xt, b, r) represent the waiting cost of class i patients

and the average time spent in the system by class i patients, respectively, as a function

of the service rate of the E/T rooms ~t in period t, patient arrival rates At in period t,

maximum number of patients allowed in the facility b and E/T room allocation r. Also,

we let cj (xjt-1, xjt) denote the cost of modifying the configuration of type j team from

period t 1 to t and oy (xjt) denote the cost of employing the personnel necessary for the

chosen configuration for team type j in period t. Assuming that all acquired additional

personnel capacity is available and becomes effective in the same period, the HCTSCP

problem can be formulated as a non-linear binary integer programming problem as follows:



i= 1 t= 1 j= 1 t= 1 j= 1 t= 1
subject to

xjkt= 1 j, t(3-2)


-~ Oij 3k kt = 0 i j (33)

si(#t, Xt, b, r)l < a V, (3-4)










cy (xt-1,xat)< TeVt (3-5)
j= 1
Zjkt E {0, 1} j k (3-6)


The objective function (3-1) minimizes the sum of the cost associated with the

time that the patients spend in the system, the cost of modifying team configurations to

change the service capacity of the teams, i.e., service rates of the associated E/T rooms,

and the cost of employing the personnel necessary for the selected team configuration.

Constraints (3-2) stipulate that one configuration must be selected for each team type in

each planning period. Constraints (3-3) assign the service rates for E/T rooms according

to the selected configurations for each team type with respect to different patient classes.

Note that (12t = 0 for t = 1,..., T due to 012k = 0 for k = 1,... ,K2. COllStraintS (34)

impose upper limits on the average time that patients spend in the system for each patient

class. Constraints (3-5) limits the amount of funds that can be allocated to changing

team configuration in each planning period. Finally, constraints (3-6) and (3-7) ensure

the integrality of team configuration and non-negativity of service rate decision variables,

respectively.

HCTSCP is a difficult non-linear binary integer programming problem with non-linear

constraints. Note that, however, HCTSCP can be represented by a T + 1-partite network,

where each l~i-;r in the network represents a time period t = 0,..., T in the planning

horizon. Let (kl, k2, t) denote the facility when configuration kgi for type j team is used

in period t. L ... rrt = 0 include a single node (kl, k2, 0), which denotes the initial

configurations for type 1 and type 2 teams. A superficial source node S is connected

to node (kl, k2,0O) only with zero arc cost. Each 1...- rt = 1,..., T contains K1 x K2

nodes, each of which represents a feasible pair of team configurations for the two teams in

period t. The cost of the arcs connecting a node (kl, k2, t 1) in periOd t 1 to a node

(k1', k2 t) in periOd t is given by C1, et (#t, Xt, b, r) + CE= cy (xjt-1, xjt) + CE~ oj (xjt),

where Zjk/~t = 1 for j = 1, 2, for t = 1, .. ., T. However, for a given node (kl, k2,ti











the upper limit on either the expected patients' time in system for any patient class or

the cost of modifying the team configurations is violated, i.e., either constraints (3-4) or

constraints (3-5) is violated, then the cost of the incoming arcs to this node are set to

M~, where M~ is a very large number. Finally, each node in 1 e. rrt = T is connected to

a superficial sink node D only with zero arc cost. Figure 3-1 provides an example of the

network representation for HCTSCP for K1 = 3, K2 = 3, and T = 2. For each team type

(al, a2) represents a configuration where the number of type -e personnel in the team is

ae for e = 1, 2. In this figure, a path from the superficial source node to S the superficial

sink node D represents a team capacity plan over the planning horizon. The HCTSCP

problem finds a capacity plan with minimum cost, if the shortest path on this graph does

not contain any arc with cost M~. Otherwise, the problem is infeasible. In the next section,

we explain how we obtain the average time in system for each patient class.


(2,4) (1,), 1(2,4), (1,1), 2

(2,4), (1,2), 1 (2,4), (1,2), 2


S ((2,4), (2,2), 1 (2,4), (2,2), 2


(2,5) (1,), O(2,5), (1,2), 1 (2,5), (1,2), 2D


(2,5), (2,2), 2


(3,4), (1,2), 1 (3,4), (1,2), 2

(3,4) (2,), 1(3,4, (22),

Figure 3-1. An illustration of the network representation for HCTSCP

3.4 Queueing Analysis

In the development and analysis of the underlying queueing models, the E/T rooms,

rather than health care teams, are viewed as the servers because a team may serve more

than one patient simultaneously, while each E/T room can be occupied by only one

patient at a time. The service of a patient is considered to begin once the patient enters

a server, i.e., an E/T room, even though all the members of the associated team may be










busy with other patients and the patient may have to wait in the room. To analyze the

time that each patient class spends in the system, we assume that the patient arrivals in

both classes are Poisson processes, and the service times for the patients are exponentially

distributed with patient-class- and team-type-dependent service rates. Given the service

rate of each E/T room type for each patient class, we can represent such a health care

system by a continuous time Markov chain (CT31C) model. We develop a DBA method

to estimate the average time that the class 1 and class 2 patients spend in a health care

facility for the preemptive and the non-preemptive cases.

3.4.1 Preemptive Case: Emergency Medicine Services

The evolution of the ED in a given period t of the planning horizon can be represented

by a CT31C. To simplify the notation, the time index t is ignored in this section. To

characterize the status of the system, we need to know not only the number of class i

patients in the system (health care facility), denoted by ni for i = 1, 2, but also the

number of class 2 patients treated in the type 1 E/T rooms, denoted by m. Therefore, we

use a triplet (nl, n2, m) to represent the state of the system, and the associated state space

is defined by

S, = { (nl, n2, m) : 0 < ni < bl, O < n2 < b2, (r1 ni1 A 2n ra2 I 1(r ni1 82 -a)

Note that because class 1 patients have preemptive priority, if the number of class 1

patients is not less than the number of type 1 E/T rooms, there is no class 2 patient being

served in the type 1 E/T rooms, i.e., m = 0 when rl < nl.

The transition rate out of state (nl, n2, m) E Sp is the sum of arrival and departure

rates of both patient classes, that is, Atl(ni < bi) + Xa2 82 < b2) 1 [n 1 11 (r0li 21~

[(n2 ) 72 22~~. The transitions entering state (nl, n2, m) arT Summarized in Table 3-1.

Using the lexicographical order sequence for the states (nl, n2), the infinitesimal

generator Q of the CTMC described above has the form of a nonhomogeneous quasi-

birth-death (QBD) process and has the stationary probability of matrix-product form

iT,1 = jT,,_lRnl, where vector xr, represents the stationary probabilities of the states









Table 3-1. The possible transitions enter state (n n2, m) for the ED setting


Event
1) A class 1 patient arrives to the system
2) A class 1 patient arrives to the system and
preempts a class 2 patient served in a type 1
E/T room
3) A class 2 patient arrives to the system, and
is admitted immediately to a type 2 E/T
room or joins the queue of class 2 patients
4) A class 2 patient arrives to the system, and
is admitted immediately to a type 1 E/T
room
5) A class-1-patient departure results in either
freeing a type 1 E/T room or starting the
service of a waiting class 1 patient
6) A class-1-patient departure results in
starting the service of a waiting class 2
patient
7) A class-2patient departure from a type 1
E/T room results in starting the service of a
waiting class 2 patient
8) A class-2-patient departure from a type 2
E/T room results in starting the service of a
waiting class 2 patient
9) A class-2-patient departure from a type 1
team results in freeing a type 1 server


Transition probability


X1P(ny
AzP(ni
ri)


1, n2, 1~~ > 0)
1, n82 1~~l > 0)I(n +m


XA2 81, n2 2, m0~a>m


XA2 81, 2 1, m 1)I(m > 0)I(n +m <
rl)I(n2 m = 2a

[(n + 1) A rl] lP(ni +l 1, m82 1 < b )


(ni + 1) liP(ni + 1, n2,m 1~~n
bl)I(m > 0)I(ni + m = ri)I(n2 2ra

m~~Pn~~ 21 1 2 2 < b2)~
0)I(nl + m = r )I(n2 2 ~

[(n2 +2 22> 1~alaa~l 2a 21 ~i
b2)

(m + 1) 21P~ln 1 l82 2 ~~n
b2)~ < 1l


with first dimension equal to nl [75]. Although the stationary probabilities of such a

process can be obtained through matrix-analytic methods, our goal is to develop fast

approximation methods to obtain the average time in system for each patient class with

sufficient accuracy. To estimate the stationary probabilities, we develop a DBA method

with the following steps: (1) decompose the state space into three submodels, for each of

which the stationary probabilities can be computed easily, and (2) combine the results of

the submodels using the linking probabilities to obtain the stationary probabilities for the

original CTMC and compute the average time in system for each patient class.

Submodels. The state space of the CTMiC for the preemptive case can be

decomposed into three submodels. Let NI~ and N2~ be random variables that denote

the number of class 1 and class 2 patients in system, respectively. The first submodel










determines the stationary probabilities of the number of class 1 patients in the system, NT~.

To determine the stationary probabilities of the number of class 2 patients in the system,

N2~, the state space S, is decomposed with respected to the first dimension by NI~ > rl and

NI~ = nl for nl < rl. This decomposition procedure forms the second and third submodels,

which are used to determine the stationary probabilities in regions NI~ > rl and NI~ = nl

for nl < rl, respectively, i.e., the conditional probabilities P(N2~ 82 1aV > 1) and

P(N2~ 82 1~V = 1) foT nl1 r1. The details of the submodels are described as follows:

* Submodel 1: P(NIV = ur): The arrival and service rates of class 1 patients are
independent of the value of N2~ because class 1 patients have preemptive priority over
class 2 patients. Therefore, the stationary probabilities P(NIV = ur) of the original
CTMiC can be determined by using an M/M/~rl/bl queueing system with arrival rate
At and service rate fit.

* Submodel 2: P(N2~ 82 1~V > 1): Given NI~ > rl, all type 1 E/T rooms must be
occupied by class 1 patients and no class 2 patient is served in the type 1 E/T rooms.
Therefore, the service rates of the class 2 patients are independent of the value of NI~,
and the stationary probabilities P(N2~ 82 na1 > 1) can be determined by using an
M~/M/r2/b2 queueing system with arrival rate a2 and service rate #22-

* Submodel 3: P(N2~ 82 1~V = 1) foT n1 I 71I 1V r1, all type 1 E/T rooms
must be occupied by class 1 patients or at least one type 1 E/T room is empty,
but a waiting class 2 patient can be admitted to an empty type 1 E/T room once
a class 1 patient leaves an E/T room and no class 1 patient arrives to the system.
Therefore, for NV~I the sum of the service rates of all type 2 E/T rooms, the empty type 1 E/T room,
and one possibly available type 1 E/T room which is occupied by a class 1 patient,
i.e.,ps, = (T2 82 22~~ + 2n r02 A 1r n1 olna 82 ra2 21, Where Pos is
approximated by Pos, = (nlfit + Xa2lnl 1 11+ 1 2

Linking Probabilities. The unconditional probabilities P(N2~ = 2) can be

represented by the conditional probabilities as follows:
n1 71
P(,N2~=na P82 li 2 12 1 i 71 IP(V I 1 71 2i 82 122N 11)P~C 1 n1
ni=0

Note that the linking probabilities, P(NIV > rl) and P(NIV = n ) for nl < rl can

be obtained from the results in submodel 1. If we know the stationary probabilities

P(NIV = nl) and P(N2~ = 2), We can calculate the expected number of patients of each









class accordingly. Then, by applying Little's formula, we can obtain the average time in

the system for each class 1 and class 2 patients.

3.4.2 Non-Preemptive Case: Outpatient Clinic Services

The evolution of the OC in a given period t of the planning horizon can also be

represented by a CT31C. As before, we again ignore the time index t to simplify the

notation. The status of the system is also characterized by the triplet (n n2, m) defined

for the ED. However, the state space associated with the OC setting is different from that

in the ED setting and is defined by

So, = {(ni, n2, m) : 0 < ni < bl, O < n2 < b2, (1 ni1 A 2n ra2 2 71 n i)

Note that if all type 2 E/T rooms are occupied, i.e., n2 > 2, 811 arTTillg class 2 patient

can be served in an empty type 1 E/T room, i.e., rl nl > 0. Thus, there is no state

(nl, n2, m) With m less than (ri ni)t A (n2 02 -

The transition rate out of state (nl, n82 m E Sup is the sum of arrival and departure

rates of both patient classes, that is, Atl(ni < bi) + Xa2 2 < b2) 1 [n 1(r m0~ 11

ma 21 [(2 ) 72 22.aa The types of transitions entering state (nl, n2, m) are the same

as the ED setting, except that the second type of transition in Table 3-1 does not occur in

the OC setting because the class 1 patients only have non-preemptive priority over class 2

patients.

Similar to the preemptive case, the matrix Q of the CT31C for the non-preemptive

case has the form of a nonhomogeneous QBD process, for which we estimate the stationary

probabilities using the DBA method. The DBA method for the non-preemptive case is

slightly different than the one for the preemptive case and includes the following four

1! in r~ steps: (1) approximate the three-dimensional CTMC by a two-dimensional one;

(2) decompose the state space into four submodels, for each of which the stationary

probabilities can be computed easily; (3) combine the results of the submodels to

obtain the stationary probabilities for the approximate two-dimensional CT3 iC and then

compute the expected number of patients of each class; and (4) represent the system by










Class 1 Type 1
Patients I lTeam Subsystem 1



Pasins ,' TTeaem2 Subsystem 2


Figure 3-2. Two-dimensional CTMC approximation

two independent simple subsystems and use the performance measures of these simplified

subsystems to adjust the results obtained in step (3).

State-dimension Reduction. The dimension of state space can be reduced from

three to two. We can treat the OC as a combination of two interacting subsystems as

shown in Figure 3-2. Subsystem 1 contains the type 1 team, type 1 E/T rooms and the

patients that are served in the type 1 E/T rooms. Similarly, subsystem 2 contains the

type 2 team, type 2 E/T rooms and the patients that are served in the type 2 E/T rooms.

We then use a two-dimensional CTMiC with state (z, n2') to approximate the original

three-dimensional CTMC, where z is the total number of patients in subsystem 1 and

n82 is the number of class 2 patients in subsystem 2. Note that z includes the number of

class 1 and class 2 patients in subsystem 1, and n2' includes the number of class 2 patients

served in subsystem 2 only. The state space of the two-dimensional CTMC is defined by

S's, = {(z, n2') : 0 < z < rl + bl, O < ~ 82~ 72 r1) + [b2 (z bl)t]I(z > r))}.

Since the OC has patient class-dependent service rates for type 1 team, we need to

know the number of patients in each class to calculate the service rate of subsystem 1.

Let Z be a random variable that denotes the total number of patients in subsystem 1, M~

a random variable that denotes the number of class 2 patients in -11 st- i.-- h 1 and N2~ a

random variable that denotes the number of class 2 patients in subsystem 2. Similarly, let

Pc, be the probability that a patient in subsystem 1 belongs to class 2. We first assume

that given Z = z, M~ follows a binomial distribution with parameters min(z, rl) and Pc,

Then, to approximate the expected number of class 2 patients in subsystem 1 given Z = z,










E [M| Z = z], we need to know whether there is a queue in subsystem 1 since it affects the

entries of class 2 patients to -Isth-i--1.11 1! Therefore, we have two cases to analyze:

* Case 1: For 0 < z < rl, no class 1 patient waits in the system, and the next patient
entering subsystem 1 can be either a class 1 or a class 2 patient. An arriving class
2 patient can enter -Isth-i--r1 11. 1 only when all type 2 E/T rooms are occupied by
patients and no other class 2 patients wait in queue, i.e., N2' r2. Thus, the total
arrival rate in this case can be approximated by Ai+Xa2(V 2 r2|Z < rl). Therefore,
for 0 < z < rl, E[M|IZ= z] can be approximated by

E [M| Z = z] = zP,,, (3-8)

where Pc, = [Xa2(V 2 r2|Z < ri)]/[At + Xa2(V 2 r2|Z < r )]. Note that the
probability P(N2~ = 2|Z < rl) can be approximated by P(N2~ = 2|Z < rl) from
submodel 1 below.

* Case 2: For rl < z < rl + bl, all type 1 E/T rooms are occupied by patients,
and only class 1 patients can enter subsystem 1. Thus, the existing class 2 patients
in s h- i--r. 11. 1 must satisfy two conditions: (1) the service time of the existing
class 2 patients must be greater than the sum of the interarrival time of class 1
patients in queue, and (2) the existing class 2 patients must have entered -Ith-i.--1. il
1 before there is a queue of class 1 patients. Therefore, for rl < z < rl + bl, we can
approximate E [M| Z = z] by

E[M|IZ = z] = max(rlPc,, TiS, + (z bi)(1 Pc )), (3-9)

where~r Pc I[A 2Z < ri))/(At + Xa2(V 2 ra2|Z < rl))]z-rt. Note
that under some particular parameter settings, we could have E[M|IZ = z] a 0 and
E[NI|~Z = z] k because Pc,, O and Nz~ + M~ = Z, for example, when the value of
< 21 Or Z is large enough. However, the number of class 1 patients in the OC can be bl
at most. Therefore, for the state (z, n2 ) With z > bl, E[M|IZ= z] should be adjusted
by (z bl) (1 Pc,) as in equation (3-9).

Submodels. After approximating the OC by a two-dimensional CTEiC, we apply

the DBA method to estimate the stationary probabilities of the system. The state space

S',, can be decomposed with respect to the first dimension by Z < rl and Z > rl and

the second dimension by N2~ > 2 and N2' r2. Then, the DBA method uses some

well-known queueing models to determine the stationary probabilities in each region, i.e.,

conditional probabilities P(N2~' = j|Z < rl), P(Z = z|N2~ > 2), P N2~' = j|Z > rl), and

P(Z = z|N2~ I 2). The details of the submodels are described as follows:









* Submodel 1: P(N2~' = j|Z < ri): Given Z < ri, no class 2 patient waits in
-I11l-i--r. 11. 2 because at least one type 1 E/T room is available in subsystem 1.
Therefore, the value of N2~ Can Only range from 0 tO T2, Which means P(N2~' = j|Z <
rl) = 0 for T2 < j < b2, and the system can be approximated by an M/M/~r2r 72queut
with arrival rate a2 and service rate #22-

* Submodel 2: P(Z = z|N2~ > 2): Given N2~ > 2, all type 1 E/T rooms must be
occupied by patients in subsystem 1. Otherwise, the waiting class 2 patients would
overflow to subsystem 1 and be served in a type 1 E/T room. Therefore, the value
of z can only range from rl to rl + bl, which means P(Z = z|N2~ > 2) = 0 for
0 < Z < rl. This system can be viewed as an M/M/1/lrl + bl queue with arrival rate
At and state-dependent service rate #z given by

~z = (r1 E[M|IZ = z])~i 1+ E[M|Z = z]#21, (3-10)

where the value of E [M| Z = z] follows from equation (3-9).

* Submodel 3: P(N2~' = j|Z > rl): Given K > rl, all type 1 E/T rooms are
occupied by patients. The behavior of subsystem 2 under this condition can be
modeled as an M/M/~r2/b2 queueing system. For state j with j > r2, Subsystems
1 and 2 may interact. That is, a waiting class 2 patient in subsystem 2 is admitted
to a type 1 E/T room if no class 1 patient waits in -111 -i.--1 1!! 1, i.e., Z = rl.
Therefore, the service rate of state j with j > r2 is the sum of the service rates of
all type 2 servers and a possibly available type 1 server, i.e., py = r2~ 22 P,, ri,
where pe, is computed from equation (3-10) and Pm, is the probability that a
type 1 server is available for a waiting class 2 patient, and can be approximated by
Ponz = P(Z = rl|N2 72 ra)i, 2 Xll-r, 2 a X1

* Submodel 4: P(Z = z|N2~ I 2): Similar to Submodel 3, we model the behavior
of the subsystem 1 as an M/M/~r /rl + bl queueing system. For state z with z < rl,
the subsystems 1 and 2 may interact. That is, an arriving class 2 patient overflow
to subsystem 1 and receives service in an empty type 1 E/T room if all type 2 E/T
rooms are occupied by patients and no class 2 patient waits in subsystem 2, i.e.,
N2' r2. Thus, for z < rl, the arrival rate of state z is the sum of arrival rates
of class 1 and possible class 2 patients, i.e., Az = X1 + Po,,12, Where Pos, is the
probability that an arriving class 2 patient receives service in a type 1 E/T room, i.e.,
P(N2~ = |Z < rl) from submodel 1. The service rate pz can be computed by

z, = (min(z, ri) E[M|1Z= z]) ~11 + E[M|IZ= z]#21, 31

where the value of E[M|IZ = z] is computed from equation (3-8) for 0 < z < rl, and
from equation (3-9) for rl < z < rl + bl.

Linking Probabilities. The unconditional probabilities P(Z = z) and P(N2' j

can be represented by the conditional probabilities as follows:









P(Z = z) = P(Z = z|N2~ 72 2~PI~ I 2) + P(Z = z|N2~ 72 2~PI~ > 2) and
P(N2 ~ 2 )= (V' = j|Z < r )P(Z < ri) + P(N2~' = j|Z > ri)P(Z > r ).

Note that the linking probabilities, P(N2~ I 2) and P(Z < rl), can further be represented

by the conditional probabilities as follows:

P(N2~ 72 2a = 72|'I Z < r )P(Z < ri) + P(N2~ I 2|Z > r )P(Z > ri) and
P(Z < rl) = P(Z < rlV' |N T72(V~ 2 r2) + P(Z < rlV' |N 2~PI~ 2 r2 -

If we substitute P(Z > r ) = 1 P(Z < rl) and P(N2~ 72 2a = 72~'I) above, we

can solve the resulting equations to obtain:


P(N2~ I 2)= and
1- P(N2~ > |Z > r )P(Z < rl' |N 2~
P(Z < rl) = P(Z < rll|N2 72 2~PI~ 72 -~

Therefore, after computing the conditional probabilities in all the four submodels, the

unconditional probabilities, P(Z = z) and P(N2~' = j), and the expected number of

patients in subsystem 1 and 2, E[Z] and E[N2 ', TOSpectively, can be computed. Then,

E [NI] and E [N2], can be computed using the following set of equations:

rl+bl
E[M] = E[M|Z= z]P[Z:= z], (3-12)
z=0
E [N~] = max {E [Z] E [M], O }, and (3-13)
E [N2] = E [N2'] + E [M]. (3-14)

Again, by applying Little's formula, we obtain the average time in system for each class 1

and class 2 patients.

Bounds. To improve the performance of the proposed DBA method for the

non-preemptive case, we use some simple queueing systems to obtain bounds on E [Z],

E [NI], and E [N2], and then adjust the results from equations (3-12)-(3-14).

*Expected total number of patients in subsystem 1 (E[Z]): The lower and
upper bounds on E[Z] are obtained by considering two M/M/~rllrl + bl queues with
state-dependent service rates from equation (3-11) with arrival rates At and At + XA2 2
respectively. Let f2 be the fraction of class 2 patients served in the type 1 E/T rooms,









which is approximated by


f2 (I~ 2 r2, Z < rl) + P(N2' r2, Z = rl) 2
X1 + X2

SP(N2~ = 2|Z < rl)P(Z < rl) + P(N2~ = 2|Z > rl)P(Z = rl) 2.(:315)
X2 X1

Note that the first term in equation (:315) represents the probability that an arriving
class 2 patient finds all type 2 E/T rooms occupied and overflows to -Isth-i.-(h il 1
immediately. The second term represents the probability that an arriving class 2
patient waits in subsystem 2 until a type 1 E/T room becomes available. Let E[NIV]
be the upper bound of E [NI]. If E [Z] is greater than E [NI], then E [NI] and E [Af] in
equations (:312) and (:313) are decreased proportionally, i.e.,

E[NIV] E[NIV]
E [NIV]ne = E [Ni]previous and E [Af]nw = E[Af],revious
E [Z] E [Z]
Otherwise, i.e., if E [Z] is less than the upper bound of E [NI], E [NI] and E [Af] in
equations (:312) and (:313) are increased proportionally.

* Expected number of class 1 patients in the system (E[NI]~): The lower and
upper bounds on E[NIV] are obtained by considering two Af/Af/rl/bl queueing
systems with service rates fit and with arrival rates At and At + X2 2, Tespectively.

* Expected number of class 2 patients in the system (E[N2]I) The lower and
upper bounds on E [N2] are obtained by considering two multi-server queueing
systems. Let E[Y] be the total number of patients in an Af/Af/rl + T2/b2 queueing
system with arrival rate At + X2 and service rate p., where rl + T2 < b2. The service
rate y is assumed to be pL = (AT Ag /(A/1 2,wer fi h fetv
arrival rate of class i patients and is approximated by
bl+rl
A~ = I(1- P(Z =/\L D)) andl A2 A2(1 2(V = b2 -.
z=bl

The resulting E[Y] is a lower bound on the total number of patients in the OC
because class 1 patients can he served in the type 2 E/T room here. Thus, the lower
bound on E [N2] can he computed by E [Y] E [NI]. Last, the upper bound on E [N2~
is obtained by considering an Af/Af/T2/b2 queueing system with arrival rate X2 and
service rate 22*.

3.5 Computational Study

In this section, we present results from our computational study, where we first

investigate the efficiency and the accuracy of the DBA method. We then use the DBA










method in conjunction with the HCTSCP formulation to test the efficiency of our capacity

planning model in making long-term personnel planning decisions.

3.5.1 Computational Performance of the DBA Method

To assess the computational performance of the DBA method, we compare the

approximate results obtained by the DBA method with the exact results obtained by

solving the steady state equations using Gaussian Elimination (GE). In preliminary

experiments we observe that GE may require significant computational effort as the size

of the model increases. Therefore, we consider an adaptation of Gaussian Elimination

(AGE) proposed by Thorson [106]. AGE algorithm avoids unnecessary row operations

by considering the fact that Q is a banded matrix, containing a large number of zero

elements. The DBA, GE, and AGE methods are implemented using C++ programming

language, and the numerical results reported are obtained using a personal computer with

a 3.0 GHz Pentium IV processor and 1 GB RAM memory.

Our parameter choices for our computational study are based on the data collected

from a participating ED. In the base case, there are eight and two rooms allocated to type

1 and type 2 teams, respectively, i.e., (rl, r2) = (8,2); service rates for class 1 and class 2

patients in type 1 E/T room are 0.25 and 0.80 patients/hour, respectively, i.e., ( II, 2~1

=(0.25, 0.80); service rate for class 2 patients in type 2 E/T room are 0.75 patients/hour,

i.e., 422 = 0.75; and the maximum number of patients from each class allowed in system

are 20 and 20, respectively, i.e., (bl, b2) = (20,20). We note that in practice, an ED is

required by law to admit all the patients that request emergency medicine services.

Therefore, essentially, the waiting room capacity is infinite, and no arriving patient is

denied of entry to the system because of lack of waiting room capacity. However, when

there is not enough service capacity, then an arriving patient can be diverted to a sister

hospital. Therefore, for modeling purposes, we include a limit on the number of patients of

each type in the system.










In our study, we consider three experimental factors including the E/T room

allocation, service capacity, and system utilization. In the first and second experiments,

we test the impact of E/T room allocation and service rate on the accuracy of DBA,

respectively. We consider rl e {2, 4, 8} and T2 E {2, 4, 8} for E/T room allocation

in the first experiment and (411, #21) E { (0.20, 0.70), (0.25, 0.80), (0.30, 0.90) } and

22~ E {0.65, 0.75, 0.85} for service rates in the second experiment. Let p denote the system

utilization. For each scenario, we use pe { 0.6, 0.7, 0.8, 0.9} to generate four instances with

different pairs of patient arrival rates, i.e., (X1, X2), using At rlfirp and X2 = r2 22 -

In Table 3-2, we report the size of the CT31C as well as the CPU time (in seconds)

required to obtain the solution using the DBA, AGE, and GE methods. In the preemptive

case, the size of the CT31C model grows if rl or T2 illCTreSeS. In addition, an increase

in rl has a more significant effect than that in T2. Similar behavior can be observed

in the non-preemptive case, however, the impact of increasing rl on problem size in

non-preemptive case is more significant than that in the preemptive case. Furthermore,

under the same E/T room allocation, the problem size of non-preemptive case is at least

twice larger than that of preemptive case for the tested scenarios in Table 3-2. Our results

show that the time to obtain the approximate results using the DBA method is negligible

and AGE is considerably more effective than GE in obtaining the exact solution.

Table 3-2. Computational requirement of the DBA, AGE, and GE methods
Preemptive Non-Preemptive
Ti T2 State Count DBA AGE GE State Count DBA AGE GE
2 2 447 0.0 0.0 0.5 2707 0.0 0.3 10.8
2 4 453 0.0 0.0 0.6 2713 0.0 0.3 10.9
2 8 4635 0.0 0.0 0.6 2725 0.0 0.3 11.2
4 2 461 0.0 0.0 0.6 4225 0.0 0.7 36.6
4 4 481 0.0 0.0 0.7 4245 0.0 0.7 37.9
4 8 521 0.0 0.0 0.9 4285 0.0 0.7 39.8
8 2 513 0.0 0.0 0.8 6609 0.0 1.9 116.1
8 2 585 0.0 0.0 1.2 6681 0.0 2.0 129.2
8 2 729 0.0 0.0 2.3 63825 0.0 2.1 161.0
Unit of running time: Second










In Tables :3-:3 and :3-4, we report the percentage error associated with the expected

time in system for the two patient classes obtained by the DBA method (when compared

to the exact solution obtained by the AGE method) in the preemptive and non-preemptive

case, respectively. Table :3-:3 shows clearly that the average time in system for class 1

patients can he correctly computed by the DBA method because submodel 1 in Section

:3.4.1 captures the exact behavior of class 1 patients in the original CT~iC. For class 2

patients, the first experiment shows that for the instances with the same levels of rl and

p, the absolute percentage error (APE) tends to decrease aS T2 increases. For example,

for the instances with rl = 2 and p = 0.6, APE decreases aS T2 illrerOSes. In addition,

for the same levels of rl/r2 and p, APE tends to decrease as rl or T2 increases as system

utilization is low, such as the instances with rl/T2=1 and p=0.6, APE decreases as rl (or

r2) increases. In other words, for a given setting of rl or rl/T2 and p, DBA performs better

in the problems with larger T2. In the second experiment, 9 scenarios of service capacity

are tested. For the same level of ( 11, 21), APE tends to increase as 022 illrerOSes for the

instances with p=0.6, which is opposite to the results of the instances with p=0.9, where

APE tends to decrease as 022 increases. Last, both experiments show that DBA tends to

underestimate class 2 patients' expected time in the system when the system utilization

is low, i.e., p = 0.6, while overestimate class 2 patients' expected time in the system when

the system utilization is high, i.e., p = 0.9.

Table :3-4 shows the results for non-preemptive case. In the first experiment, we

observe that for the instances with the same levels of rl and p, the APE for class 2

patients tends to decrease aS T2 illrerOSes, which is the same as the preemptive case. In the

second experiment, for the same levels of ( II, 21) and p, APE tends to increase as 022

increases for both class 1 and class 2 patients. In addition, DBA tends to underestimate

class 1 patients' expected time in system while it tends to overestimate class 2 patients'

expected time in system. For class 1 patients, APE is less than 5 percent for all tested

instances, i.e., DBA yields a more reliable estimate for class 1 patients' time in system.




















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3.5.2 Computational Performance of the HCTSCP Model

To illustrate the effectiveness of the HCTSCP model used in conjunction with the

DBA method in making long-term personnel planning decisions, we compare the capacity

plans generated by the HCTSCP model using the approximate and exact average time in

system for different patient classes obtained by the DBA and AGE methods, respectively.

We also study the sensitivity of the results obtained by the HCTSCP model to several

problem parameters.

We consider the personnel planning problem over a three-year planning horizon where

the unit planning period corresponds to a quarter of a year, i.e., T = 12. We assume that,

for all t, the initial service rates for class 1 and class 2 patients are (0110, #210, 220o

(0.25, 0.80, 0.75) patient/hour, the allowable maximum numbers of patients in system

are (bl, b2) = (20,20) patients, the E/T room allocation is (rl, T2) = (8,2), and the upper

bound on the amount of time that a patient spends in the system are (caz, ag~) = (4.75,

4.00) hours. We consider a case where there are two types of personnel included in each

team, and there are six and four feasible configurations for type 1 and type 2 teams,

respectively. Table 3-5 shows the number of different types of personnel with different

skill sets included in each configuration and the corresponding service capacity as well as

operating cost for each team. Without loss of generality, we assume that patient waiting

costs increase linearly with patients' time in system, unit delay cost for each patient class

(UPit, UP~t) arT Set to ($400,$100) /hour per patient, and the funds that can be allocated

to change the service capacity of the teams in the facility 7t= $17,000 for all t. Finally, we

assume that personnel hiring or termination costs are zero.

An examination of emergency medicine practices show that patient arrivals to the

ED exhibit seasonality. We generate the total quarterly arrival rate of patients using a

seasonally adjusted trend line, represented by a function of the form Xt = 6mod(t,4)U( 0 + bt)

where be is the quarterly seasonality factor for season i (where we have the following

estimates for the seasonality factors by = 0.8, 62 = 1.0, 63 = 1.2, and 64 = 1.0), a is









Table 3-5. Team configurations
Type 1 Team Type 2 Team
kl al a2 ~11 21COSt k2 81 82 2 COSt
1 2 4 0.23 0.75 56,000 1 1 1 0.60 11,500
2 2 5 0.25 0.80 63,000 2 1 2 0.75 16,500
3 2 6 0.28 0.85 70,000 3 2 2 1.00 23,000
4 3 4 0.35 1.00 70,000 4 2 3 1.10 28,000
5 3 5 0.40 1.10 77,000
6 3 6 0.45 1.20 84,000
Unit of '. patients/hour. Unit of cost: $/quarter

a uniformly distributed random number (where we have u ~ U[0.8, 1.2]), Ao = 2, b

= 0.04, and t = 1, .. ,12. Note that we implicitly assume that the patient demand

increases linearly by 2 percent every quarter. We assume that the fraction of class 1

patient is fit = 0.85 for all t, i.e., Xlt = flt~t and X~t ( lt Xt. We note that our

parameter choices are mainly according to the characteristics of the data collected from

the ED, which is represented by the preemptive case. In order to eliminate the effects that

may be due to a specific health care facility, we use the same set of parameters for the

non-preemptive case also.

In our study, we considered three experimental factors including the fraction of class

1 patients, fit, the unit patient treatment cost for each patient class, (UPit, UP~t), and

the maximum allowable average time in system for each patient class, (cx, c82). Each

parameter is tested at three levels as listed in Table 3-6. For each experimental setting, we

generated 25 random instances of the patient arrival streams over the planning horizon.

We solved all the instances for each of the settings by using the HCTSCP model with the

AGE method and with the DBA method to compare the performance of the two methods.

Table 3-6. Parameter settings
Parameters Level 1 Level 2 Level 3
Experiment 1 flt(~ ) 80 85 90
Experiment 2 U~it ($/hour/patient) 200 400 800
UP~t ($/hour/patient) 50 100 200
Experiment 3 atl (hours) 4.50 4.75 5.00
a2~ (hours) 3.50 4.00 4.50










As explained in Section :3.3, we build a network to represent the HCTSCP problem.

To compute the patient delay cost for each are, we obtain average time in system for each

patient class approximately (exactly) using the DBA (AGE) method. We note that we

do not use the GE method in this experiment, since the AGE method is shown to be

considerably more efficient in Section :3.5.1. After we construct the network, we find the

shortest path using Dijkstra's algorithm [2]. The results show that the network for the

12-period HCTSCP problem with six and four feasible configurations for type 1 and type

2 teams, respectively, contains 290 nodes and 6,384 arcs. The average time required to

obtain the optimal capacity plan is 11.0 seconds for preemptive case and 549.8 seconds for

non-preemptive case if we use the AGE method in building the network for an instance of

the HCTSCP problem. In contrast, the DBA method requires 0.03 seconds, on average,

for both the preemptive and non-preemptive cases. Therefore, using the DBA method

in building the network for an instance of the HCTSCP problems is considerably more

efficient than using the AGE method.

In order to measure the accuracy of the DBA method, we compare the team capacity

plans generated by the two approaches based on a ;.0 .1.;<./.;, indexr. Specifically, we count

the number of periods where team configurations chosen are the same in both approaches

and then divide this counting by 12 for each care team type to determine the value of

the similarity index. The results of our comparison for each experimental factors are

summarized in Tables :3-7, :3-8, and :3-9. Cells range from 0 percent (absolutely different)

to 100 percent (perfectly similar). For instance, if a cell has a value of 75 percent, then

among 12 periods, the HCTSCP model with DBA gives the same results in 8 of them as

the HSCTSCP model with GE.

In Table :3-7, we observe that the HCTSCP model with DBA works very well in

the preemptive case. However, in the non-preemptive case, as the fraction of class 1

patients increases, the performance of the HCTSCP model with DBA deteriorates as it

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configuration with a higher service rate. This can he attributed to our earlier observation

that the DBA method overestiniates the average time in system for class 2 patients.

Table 3-8 shows that the unit patient delay cost does not impact the accuracy of the

HCTSCP model in the preeniptive case. But in the non-preeniptive case, its accuracy

in determining the team configuration of type 2 team goes down, as unit class 1 patient

delay cost or unit class 2 patient delay cost decreases, and it tends to overestimate the the

required service capacity of the type 2 team by 1 level. Table 3-9 shows that the nmaxiniun

allowable average time in system does not materially impact the accuracy of the HCTSCP

model in either the preeniptive case or the non-preeniptive one. In suninary, the HCTSCP

model with DBA is efficient and accurate in solving the capacity planning problems,

particularly in the preeniptive case, e.g., the ED application. For the non-preeniptive case,

e.g., the OC application, its accuracy in type 2 care team requirements is not as precise.

3.6 Concluding Remarks and Future Research Directions

In this paper, motivated by the emergency medicine services at a coninunity hospital

and specialty services at an outpatient clinic, we examined the health care team service

capacity planning problem. These health care systems provide services to two 1!! ri ~

patient classes using two types of health care teams and two types of E/T rooms. While

class 1 patients are served by the type 1 team in the dedicated E/T rooms only, class 2

patients can he served hv either the type 1 or the type 2 team in the E/T rooms dedicated

to the corresponding team. However, the type 1 team delivers service to a class 2 patient

only if there are no class 1 patients waiting in the system and all type 2 E/T rooms are

occupied by other class 2 patients. The service capacity of each team is measured by the

service rates of the dedicated E/T rooms for each patient class. Although this setting is

similar to some other studies in the literature, the distinguishing characteristic of our work

is that we have patient-class- and team-type-dependent service rates.

In our work, we first considered the case where the class 1 patients can preenipt class

2 patients served in type 1 E/T rooms, which is typical in emergency rooms in hospitals.










We also analyzed the non-preemptive case, common in outpatient clinic settings. We

developed queueing models for both cases, and developed approximation procedures

to estimate the average time that each patient class spends in the system. Through

an extensive computational study, we illustrated that our DBA method provides the

performance measures of interest efficiently with sufficient accuracy. Using the results

of the queueing an~ lli--- we then developed a non-linear binary integer programming

model to determine the minimal cost capacity plan of health care teams that a health

care facility should employ over a planning horizon to deliver service for the patients

while ensuring that the average time that each patient class spends in the system does not

exceed certain values. Our computational study showed that our approximation approach

provides sufficiently accurate results that can be used in practice to make long-term health

care team service capacity planning decisions.

We note that in our queueing analysis, we assumed exponentially distributed

interarrival and service times to preserve analytical tractability. Extending our work

to consider general arrival and service processes is a potential area for future research.

Moreover, in our study we analyzed systems where patients are categorized into two

classes. Although this classification is widely used in the health care industry, some clinics

and hospitals further classify each of the classes into two or more subclasses [100]. The

presence of multiple subclasses in each patient class magnifies the problem size rapidly and

cannot be solved through general numerical methods (e.g., the GE or the AGE method).

Therefore, developing approximation procedures for such settings would be of theoretical

and practical interest. In our work, we focused on the long-term capacity planning of

the health care teams, treating the E/T rooms as the servers and assuming that the

service rates of the rooms for different patient classes are given, and room allocation is

fixed over the planning horizon. For more detailed analysis, the set of tasks required for

the diagnosis, monitoring, and treatment of a patient can be modeled using a queueing










network, and the service rates of the E/T rooms under different room allocations and

team configurations can be further investigated.










CHAPTER 4
HOSPITAL BED ALLOCATION PROBLEM

4.1 Introduction

In this chapter, we introduce the hospital bed allocation (HBA) problem. The HBA

problem is an extension of the AHBCP problem in C'!. Ilter 2, which is concerned with

determining the optimal .I__-oegate bed capacity plan over a finite planning horizon. After

the .I_ gate bed capacity is specified, the next step involved is concerned with the

allocation of .l__-oegate bed capacity among different medical care units (j!CI~s) (e.g.,

neurosurgery, oncology, pediatrics, etc.).

Ineffective allocation of existing bed capacity among different medical service units

can lead to service quality problems for the patients along with operational and/or

financial inefficiencies for the hospitals. An arriving patient may be declined or placed

on hold by an MCIT, if there is no bed available to accommodate the patient in the unit.

In this case, the patient may be subject to same health risks due to the necessity to

find an alternative health care provider or wait until a bed becomes available. From the

hospital's perspective, the potential revenue is either lost or deferred, which may lead to

some financial inefficiencies. Similarly, an arriving patient may be accepted for treatment

but can he accommodated in another MCIT (e.g., an arriving obstetrics patient can he

boarded in neurosurgery). In this case, the patient may be subject to same unnecessary

health risks due to the necessity to be boarded together with patients with more different,

possibly more serious, health conditions. Fr-om the hospital's perspective, the potential

revenue is collected but the operational costs may increase, as the patient may be boarded

in an MCIT where the service resources are more expensive (e.g., neurosurgery nurses

may have additional qualifications than obstetrics nurses and the medical equipment in a

neurosurgery department are typically more expensive than the ones in obstetrics).

To effectively utilize existing bed capacity, hospital administration could choose from

among a number of alternative planning strategies to to find an allocation of existing










.I_ egate bed capacity among different MCUs. In particular, there are four practical

planning strategies, including

* the expected bed occupancy is balanced across entire hospital,
* the expected net profit of the hospital is maximized,
* the occurrence of bed shortages is minimized, or
* the number of patients rejected is minimized.

In this work, we focus on the first strategy and develop a mathematical programming

formulation to address this problem. We also develop effective solution approaches to

obtain high quality solutions particularly for large-sized, realistic test instances.

The remainder of this chapter is organized as follows. In Section 4.2, we review the

related literature. Section 4.3 presents mathematical programming formulation for HBA.

We develop three heuristic solution approaches in Section 4.4. In Section 4.5, we present

results from our computational study that evaluates the computational performance of the

approaches developed in Section 4.4. Section 4.6 includes a discussion of the results and

-II_ -- -r- future research directions.

4.2 Literature Review

Most health care managers apply relatively simple approaches, such as the use of

target occupancy level with average length of stay, to forecast bed capacity required for a

hospital or an MCU. Yet, the failure to adequately consider the uncertainties associated

with patient arrivals and time needed to treat patients by using such simple approaches

may result in bed capacity configurations where a large portion of patients may have to

be turned .l.--li- [46]. To take the stochastic nature of health care systems into account,

researchers utilize queuing and simulation models in determine the appropriate bed

capacity configuration.

The application of queueing theory allows for the evaluation of the expected

(long-run) performance measure of a system by solving the associated set of flow balance

equations. Mackay and Lee [80] evaluate the choice of models for forecasting bed capacity

and -II__- -r to use compartmental flow model, which models patient flow through a










hospital as flow through a sequence of compartments. Patients with short length of stay

may leave the system after visiting the first compartment, otherwise, patients move

to the next compartment until their length of stay are reached. The benefit of using

compartmental flow model is that it can capture the variation in bed occupancy without

using a sophisticated method.

Gorunescu et al. [43] model a department of geriatric medicine as an Af/Af/c/K

queueing system to investigate the interrelationships between admission rates, length

of stay, number of allocated beds, and probability that an arriving patient is denied

admission. K~ao and Tung [63] present an approach for allocating beds to care units in

a hospital to minimize the expected patient overflow, i.e., the rate at which patients are

denied admission due to inavailability of bed capacity. They model each service as an

Af/G/oc queueing system and use normal approximation in computing number of patient

overflows. The bed allocation problem is solved in two stages. The first stage distributes

the 1 in 4 Gly of beds such that no gross imbalances in bed utilization among all care

units are observed and a pre-specified fraction of patients can stay in the units designated

for the unit. The second stage uses marginal analysis to allocate the remaining beds to

minimize the expected total patient overflow.

Discrete-event simulation is useful for the analysis of systems with complex behavior,

including health care systems. Discrete-event simulation has been widely applied in

health care services [58] to study the interrelationships between admission rates, hospital

occupancy, and several different policies for allocating beds to AICUs. Harrison et al.

[52] construct a simulation model where patients' stay in hospitals are classified into

three stages, which represent different phases of care provided. The output obtained from

the model matches the mean and the variability associated with actual bed occupancy

data. The model is used to identify daily occupancy distributions, study trade-offs

between overflow and bed capacity levels, and investigate the effects of various changes.

Akkerman and K~nip [6] use Markov chain approach to specify the number of beds needed










for two hospital wards, and then utilize discrete-event-simulation to obtain detailed

information on expected bed occupancy and patient rejection levels. Masterson et al. [84]

use discrete-event simulation to investigate the interrelationships between bed occupancy,

average number of patient deferred, and different bed allocation and operating policies.

Harper [51] develops a simulation model for the planning and management of hospital

beds, operating theaters, and workforce needs. The model captures the complexity

of health care systems by incorporating the variability for each patient group such as

monthly, daily, and hourly demand as well as the distributions of length of stay and

operation times. K~im et al. [68] analyze an intensive care unit with 14 beds and develop

a simulation model to evaluate different bed-reservation schemes to reduce the number of

cancelled surgeries.

Some papers consider the hierarchical relation between care units. For example, after

a mother-to-be delivers her child in the labor and delivery unit, she should be moved to

the postpartum unit for recovery. When the capacity downstream is insufficient, patients

are forced to stay at the current care units with typically more expensive equipment

blocking the capacity at these upstream care units. To take the interactions among care

units in a hospital into account, Cochran and Bharti [30] first apply queueing network

methodology (without blocking) to find a balanced bed allocation, which is obtained

through trial-and-error work. Then, they use simulation analysis to estimate the blocking

behavior and patient sojourn times. Galviio et al. [39, 40] apply a three-level hierarchical,

capacitated model to determine the capacity required in perinatal health care facilities,

which is categorized into three levels: basic units, maternity homes, and neonatal clinics

where intensive care unit for babies is available.

In our work, we utilize a different approach by integrating results from queueing

theory into an optimization framework. Specifically, we model each MCU in a hospital

as an M/M/~c/c queueing system to estimate the probability of rejection when there are

c beds in the unit. We then develop an optimization model to allocate the .I__oregate bed










capacity across different MCUs. The purpose of this work is to develop efficient solution

approaches to solve this problem.

4.3 Problem Formulation

We now begin the mathematical formulation with the objective of balancing bed

occupancy throughout the hospital. We consider a hospital with D MCUs and B beds

available. For each service there are lower and upper limits on the number of beds

allocated, denoted by 14 and ui, respectively. In addition, there is a lower bound on the

bed occupancy for each MCU, denoted by y. To analyze the bed occupancy of each MCU,

denoted by pi, we assume that the patient arrivals of each MCU are Poisson processes

and the lengths of stay at each MCU are exponentially distributed. We assume that an

arriving patient to MCU i is rejected, if all beds designated for the MCU i are occupied.

Let pi denote the probability of rejecting an arriving patient of MCU i, and xi be the

number of beds allocated to MCU i. Let p be the bed occupancy of the entire hospital, As

be the patient arrival rate and 1/ps be the average lengths of stay at MCU i. Let xi be the

decision variables of HBA problem, i.e., the number of bed allocated at MCU i. We then

formulate the HBA problem as a nonlinear integer programming formulation as follows:


min |p p| (4-1)
i= 1
subject to

x4=B (4-2)
i= 1


xi! m!
m=o
(1 pi) As
pi = Vs (4-4)


p = (4-5)
i= 1
le < xi < ai Vi4-6
< < i 1 Vi4-7
xi E Z* Vi(4-8)










The objective function (4-1) minimizes the total deviation of the bed occupancy for

each of the MCUs from the overall average bed occupancy for the hospital. Constraint

(4-2) limits the total number of beds allocated among the MCUs to the number of total

beds available in the hospital. Constraint set (4-3) represents the probability of rejecting

patients arriving to service i as a function of arrival rate As, service rate ps and bed

capacity xi for service i. Constraint set (4-4) represents the bed occupancy of the service i

as a function of effective arrival rate (1 pi)As, service rate ps and bed capacity xi for each

service i. Constraint (4-5) specifies the overall average bed occupancy for the hospital.

Constraint sets (4-6) and (4-7) impose the lower and upper bounds on the bed capacity

and the bed occupancy for each service i, respectively. Finally, constraint (4-8) ensures

that the decision variables are non-negative integers.

HBA is a difficult nonlinear binary integer programming problem with nonlinear

constraints. Note that constraint set (4-3) involves the factorial function on xi and the

summration of the termn~I fr~omn m = 1 up to m11 = xsl w~hic~h increases the c~omplexuity

of the problem significantly. Since decision variables xi's assume discrete values, we

can introduce the splitting variable yij, where yij takes value of one if xi = bij and zero

otherwise, where bij E {li, 14 + 1, ..., as 1 ui}. Let pij and pij represent the bed occupancy

and probability of rejecting an arriving patient of MCU i, respectively, when bed capacity

of MCU i equals bi.

As a result, we can obtain an equivalent linear binary integer programming problem

that can be stated as follows:

D ug -14
minl boys;V (4 9)
i= 1 j= 1
subject to
D u -1
b oysZ,i = B (4-10)
i= 1 j= 1










pi (Ag/ps)bqiC (i in"
p = bii! m!= V i, (4-11)

pij = V j(4-12)
D ug -14
p = (4-13)
i= 1 j= 1
< I-
y =1 i(4-15)
j= 1
sij > Pij P i (4-16)
sij > P payV (4-17)
yij E {0, 1} i (4-18)
be, {ls14+1...as 1us} i~j(4-19)

We have four sets of additional constraints. Constraint set (4-15) ensures that

only one splitting variable takes the value of one. Constraint sets (4-16) and (4-17)

compute the deviation of the bed occupancy for each service i from the overall average bed

occupancy for the hospital. Constraint set (4-18) ensures that the decision variables take

binary values. Finally, constraint set (4-19) provides information of the discrete options of



4.4 Solution Algorithms

The HBA problem with variable splitting formulation can be solved by commercial

MILP solvers, however, it takes time to obtain the optimal solutions of real-size problems.

This section describes three solution approaches we develop for the HBA problem, which

includes genetic algorithm (GA), greedy randomized adaptive search procedure (GRASP)

and a hybridization of GA and GRASP.

4.4.1 Genetic Algorithm

GA constructs a population of solutions and generates new generations of solutions

mimicking the behavior of population genetics [41, 53]. In particular, two members of the

current population of solutions are chosen randomly and used to generate new offspring










which are then retained for the next generation of solutions if they qualify. Figure 4-1

depicts the pseudo-code of the GA. A preprocessing step ensures that a test instance is

feasible by verifying the validity of the following inequalities:


le i< B (4-20)
i= 1

us > B(4-21)
i= 1
p(14) > 7 V (4-22)

where p(14) represents the bed occupancy with bed capacity 14 of service i.

procedure GeneticAlgorithm(instance)
1Read and preprocess the input data;
2 Solution* <- 0;
3 Population <- GeneratePopulations (PopulationSize);
4 NumGenNotImprove <- 0;
5 for m = 1 to MaxNumGenerations do
6 Children <- ProduceOffspring(Population, NumGenNotImprove);
7 Population <- UpdatePopulation(Population, Children);
8 Solution* <- Up dateSolution( Solution*, Children, NumGenNot Improve);
9 end for
end GeneticAlgforithm.

Figure 4-1. Pseudo-code of the genetic algorithm

GA starts by generating a set of distinct feasible solutions to form an initial

population of solutions. These solutions are generated through an occupancy-driven

approach, and then adjusted to feasible by a random-rectified approach (see Figure 4-2).

The occupancy-driven approach first treats each MCU as an M/M/cl queueing

system, and then allocates a number of beds to the MCU such that the bed occupancy is

close to y (see Figure 4-3). As it can be observed from constraint set (4-3), specifying the

probability of rejecting a patient from an MCU that is modeled as an M/M/~c/c queueing

system requires the knowledge of the number of beds allocated to that particular MCU.

Since our objective is to generate some solutions for the initial population of GA, rather









procedure GeneratePopulations (PopulationSize)
1 Population <- 0);
2 while NumlnSet (Populaiton) < PopulationSize do
3 x <- OccupancyDrivenAllocation(A, p, y);
4 x <- RandomRectified( x, 1, u, y);
5 if x Sf Population then
6 Population <- Population U x;
7 end if
8 end while
9 return (Population);
end GeneratePopulations.

Figure 4-2. Pseudo-code of the population generating procedure

than spending time to find the corresponding bed capacity for an M/M/~c/c queueing

system, we model the MCU as an M/M/cl queueing system to initialize the bed capacity.

procedure OccupancyDrivenAllocation(A, p, y)
1 x <- 0;
2 for i = 1 to D do

4 end for
5 return (x);
e nd O ccup ancyD rivenAllo cation.

Figure 4-3. Pseudo-code of occupancy-driven allocation

The solutions generated from the occupancy-driven approach may not satisfy

constraint sets (4-2), (4-6) and (4-7), i.e., constraint sets associated with the total

number of beds available, lower and upper bounds and bed occupancy (of an M/M/~c/c

queueing system). We use a random-rectified procedure to fine-tune the initial bed

allocation. Figure 4-4 shows the pseudo-code of the random-rectified procedure. We

first revise the bed capacity of each MCU such that the constraints (4-6) and (4-7) are

satisfied. Then, when the total number of beds allocated is greater (less) than B, an MCU

is chosen randomly and its bed capacity is decreased (increased) by one, if this operation

does not violate constraint (4-6). The step of choosing an MCU randomly and adjusting

its capacity accordingly is repeated until the total number of beds allocated is equal to the

overall bed capacity available, B.









procedure RandomRectified( x, 1, u, y)
1 for i = 1 to D do
2 while p(xi) < y do
3 xi <- xi 1;
4 end while
5 xi <- min~xi, ui};
6 xi <- max~xi,14}>;
7 end for
8 while NumBedAllocated(x) < B do
9 i <- RandomlySelectInteger(D);
10 xi --min~xi +1i, u ;
11 end while
12 while NumBedAllocated(x) > B do
13 i <- RandomlySelectInteger(D);
14 xi <- max~x { x, 1i,1 }
15 end while
16 return (x);
end RandomRectified.

Figure 4-4. Pseudo-code of random-rectified procedure

After the initial population is formed, GA starts to produce offspring. Two types of

genetic operators are considered in this work; one is simple single-point crossover, and

the other is mutation. In simple single-point crossover (see Figure 4-5), we first randomly

select two parents from the population, and then swap bed capacity between the randomly

selected [D/2] MCUs. That is, if the MCU i is selected, the bed capacity of MCU i of

the first parent's chromosome is swapped with the bed capacity of MCU i of the second

parent's chromosome. The resulting solutions are the offspring solutions produced by the

selected parent solutions. Figure 4-6 shows an example with four MCUs and 100 beds,

where the second and the third MCUs are chosen to swap their bed capacity. Note that

the crossover may produced an offspring which does not have a chromosome specifying a

feasible solution. In this case, the random-rectified approach is emploi- a to adjust the bed

capacity allocation.

The second genetic operator, mutation, is invoked if the current best solution is not

improved throughout the evolution of a pre-specified number of consecutive generations.

The mutation employs the occupancy-driven approach with the average occupancy









procedure Crossover(Population)
Sx <- RandomlySelectElement (Population);
2 y RandomlySelectElement (Population);
3 while y= x do
4 y <- RandomlySelectElement (Population);
5 end while
6 SwapSet <- 0);
7 while NumlnSet(SwapSet) / [D/2] do
8 i <- RandomlySelectInteger(D);
9 if i ( SwapSet then
10 SwapSet <- SwapSet U i;
11 end if
12 end while
13 for i = 1 to D do
14 if i E SwapSet then
15 temp <- xi;
16 xi <- ys~;
17 ye~ <- temp;
18 end if
19 end for
20 x <- RandomRectified(x);
21 y <- RandomRectified(y);
22 if f(x) < fly) then
23 return (x);
24 else
25 return (y);
26 end if
end Crossover.

Figure 4-5. Pseudo-code of crossover procedure

(10, 20, 30, 40) (10, 16, 24, 40)

(38, 16, 24, 22) (38, 20, 30, 22)

Figure 4-6. Example of crossover

of the current best solution to produce an offspring. Again, this mutated offspring is

adjusted through the random-rectified procedure to ensure feasibility. Then, the value of

NumGenNotImprove is set to zero.

If the offspring produced from either crossover or mutation procedure does not exist

in population and is better than the current worst solution in population, then the current









procedure Mutation(x*)
1 p* p (x*);
2 x OccupancyDrivenAllocation(A, p, p*);
3 x RandomRectified( x, 1, u, y);
4 NumGenNotImprove<- 0;
5 return (x);
end Mutation.

Figure 4-7. Pseudo-code of the mutation procedure

worst solution in the population is replaced with the offspring to form a new generation

with the other existing solutions. Otherwise, i.e., either the offspring exists in population

already or the offspring is not better than the current worst solution in the population,

this offspring is ignored and the next offspring is produced using the procedure described

above. Note that the current best solution is also updated if the new offspring has a better

objective function value than that of the current best.

4.4.2 Greedy Randomized Adaptive Search Procedure

Greedy randomized adaptive search procedure is a multi-start approach [36] that is

widely used for combinatorial optimization problems. Each GRASP iteration consists of

two phases: construction and local search. Figure 4-8 depicts the pseudo-code for GRASP.

The construction phase creates a feasible solution, whose neighborhood is explored by

the local search phase to find a locally optimal solution. The algorithm stops after a

pre-specified number of iterations is executed, which is denoted by MaxNumIterations in

Figure 4-8.

procedure GRASP(instance)
1 Read and preprocess the input data;
2 Solution* <- 0;
3 for n = 1 to MaxNumIterations do
4 Solution <- GreedyRandomizedConstruction (a);
5 Solution <- LocalSearch (Solution);
6 Solution* <- Up dateSolution( Solution, Solution*);
7 end for
end GRASP.

Figure 4-8. Pseudo-code of GRASP










The construction phase builds a feasible solution in a greedy manner. First, each

MCU is allocated an initial bed capacity at its lower bound to form an incomplete

solution. Then, one bed is added to an MCU if this capacity expansion does not destroy

the upper bound constraint and the increment on the objective value is at an acceptable

level. The pseudo-code of the construction procedure is illustrated in Figure 4-9, where

the ej in line 22 represents a zero vector except the jth element equals one. Lines 15

through 20 in Figure 4-9 build a restricted candidate list (RCL), which records the set

of MCUs for which adding one more bed to the MCU does not violate the feasibility and

has the potential to improve the objective function value at an acceptable level. The

threshold of increment on the objective function value is controlled by the parameter

as {0(, 1} on line 17 in Figure 4-9. An MCU is included in RCL, if adding one more bed

to the unit is feasible, and the increment on the objective function value is not greater

than 6min + a~(6max 6min), where 6min and 6max represent the minimum and maximum

increments on objective function value after incorporating one more bed to current

solution, respectively. Note that the lower the a~ is, the greedier the procedure is. A bed is

added to an MCU selected randomly from RCL, until the total number of beds allocated

equals B.

During the local search phase, the neighborhood of the feasible solution created in the

construction phase is fully investigated to find the local optimum. A neighbor solution is

produced by swapping one bed from one MCU to another, if this swap does not destroy

the feasibility of the solution. The pseudo-code of the local search procedure is illustrated

in Figure 4-10. Here, we adopt the best-improving strategy that is we evaluate all feasible

neighbors and choose the neighbor that improves the objective function value the most.

4.4.3 Hybridization of GA & GRASP

We develop a hybridization of GA and GA. The pseudo-code for the hybrid approach

(HA) is given in Figure 4-11. On each iteration of the HA, a set of elite solutions is

generated, in contrast to a single initial solution used by GRASP. Then, an offspring is










procedures GreedyRandomizedConstruction(a~)
1 x <- 0;
2 for i = 1 to D do
3 xi = li;
4 end for
5 while NumBedAllocated(x)/ B do
6 6min t- +oo;

8 for i = 1 to D do
9 if xi + 1 < ui then
10 si <- f (x+ei) f (x);
11 6min <--min {6min, be };
12 6maz <- max {6max,6 i);
13 end if
14 end for
15 RCL = 0);
16 for i = 1 to D do
17 if xi + 1 < ui and be < 6min + a~(6max 6min) then
18 RCL <- RCL U i;
19 end if
20 end for
21 j <- RandomlySelectElement (RCL);
22 x <- x + ej;
23 end while
24 return (x);
end GreedyRandomizedConstruction.

Figure 4-9. Pseudo-code of greedy randomized construction procedure

produced using the crossover procedure of GA and improved by the local search procedure

of GRASP. The previous two steps are repeated, until a pre-specified number of iterations

are completed. The characteristic of the HA is that on each iteration the threshold a~ for

the greedy randomized construction phase is updated by a decreasing function p(u), i.e.,

a~ decreases as the number of iterations, n, increases. In this way, the HA can test more

than one setting of a~ and reduce the probability of converging to a local optimum solution

prematurely.

To form an elite set of solutions, we first use the GreedyRandomizedConstruction

procedure of GRASP to generate NumGRASP distinct solutions, and then choose the best

ESize of them, where NumGRASP > ESize. The pseudo-code is presented in Figure 4-12.










procedure LocalSearch(x)
1 x*<-x;
2 f*<- f (x);
3 for i = 1 to D do
4 for j = 1 to D do
5 if i / j, xi 1 > 14, xi + 1 I ui and p(xi + 1) > y then
6 if f (x-ei + ej) < f then
7 x*<-x-ei + ej;
8 f*<- f (x-ei + ej);
9 end if
10 end if
11 end for
12 end for
13 return (x*);
end LocalSearch.

Figure 4-10. Pseudo-code of local search procedure

procedure Hybrid(instance)
1Read and preprocess the input data;
2 Solution* <- 0;
3 for n = 1 to MaxNumIterations do
4 ~ <--p (u);
5 EliteSet <- GenerateEliteSet(a~, Solution*);
6 for m = 1 to MaxNumGenerations do
7 Solution <- Crossover(EliteSet);
8 Solution <- LocalSearch( Solution);
9 Solution*<- UpdateSolution(Solution, Solution*);
10 end for
11 end for
end Hybrid.

Figure 4-11. Pseudo-code of HA

4.5 Computational Study

In this section, we present results from our computational study, where we investigate

the computational efficiency of the proposed solution approaches. Specifically, we measure

the efficiency of the approach using the CPU time needed to obtain the solution and

relative error in the objective function value, given by

IA f*









procedure GenerateEliteSet(a~, Solution*)
1 EliteSet <- 0);
2 for j = 1 to NumGRASP do
3 x <- G reedy Randomiz ed Construct ion ( a);
4 while xe EliteSet do
5 x <- GreedyRandomizedConstruction(a~);
6; end while
7 y <- the worst solution in EliteSet;
8 if f(x) < fly) then
9 EliteSet <- EliteSet \ y U x;
10 end if
11 Solution*<- UpdateSolution(x, Solution*);
12 end for
13 return (EliteSet);
end GenerateEliteSet.

Figure 4-12. Pseudo-code of elite set generation procedure

where fA denotes the objective function value obtained by using approach A, where

AE {GA, GRASP, HA}.

In our study, we consider three settings each with different instance sizes to evaluate

the impact of the size of the instance on the performance of the proposed solution

approaches. The problem size is varied with the number of total beds available in the

hospital, B, and number of MCUs in the hospital, D. For the total number of beds and

the number MCUs available in the hospital, we consider three levels that correspond to

small, medium, and large sized hospitals. In particular, we consider hospitals with 750,

1,000 or 1,250 beds and 30, 40 or 50 MCUs. For each setting, we generate 30 random

instances. Each random instance is obtained by generating parameters that correspond

to patient arrival and service rates along with lower and upper bounds on the number of

beds available in each MCU. Specifically, each parameter is obtained by using the formula:

(mean value)+u, where u is drawn from the distribution U [0.4, 1.6]. The mean values of

the random parameters are listed as follows:

1. Mean arrival rate (Ag): 10 persons/per unit time for each MCU i;
2. Mean service rate (ps): 0.5 persons/per unit time for each MCU i;
3. Mean lower bound (14): 10 beds for each MCU i; and










4. Mean upper bound (ui): 80 beds for each MCU i.

Last, the minimum bed occupancy, y, is 'TI' for all MCU all problems. The

experiment is implemented on a workstation with two Pentium 4 3.2 GHz processor

and 6 GB of memory.

To obtain a near-optimal solution as a comparison basis, we use CPLEX to obtain

the optimal solutions for the test instances. For each test instance, CPLEX is stopped

if the relative stopping tolerance of 0.01 is satisfied, or CPU time of 3,600 seconds is

used. Table 4-1 reports the results obtained by CPLEX. As we expect, the number of

instances for which CPLEX can determine and verify the optimal solution within one

hour of CPU time decreases as the problem size increases. Moreover, for the instances

with small problem size, CPLEX takes more than 1,800 seconds on average to solve one

instance.

4.5.1 Summary of Results Obtained by GA

GA is executed for 200D generations and performs a mutation when best solution

is not improved for D generations, where D is the number of MCUs in consideration. To

evaluate the impact of the population size, we consider three levels of D, 2D and 3D for

each setting. Table 4-2 reports the results of GA, where the information is classified into

two lI-;-ir The first by. ;-r distinguishes the instances with different sizes, and the second

specifies the parameter setting of GA, including number of generations and population

size. Note that if the relative error of an instance is less than zero, it is replaced by zero

when computing the average relative error of the corresponding instance.

In general, GA performs quite well in terms of average relative error and CPU time.

For any of the 90 instances, GA spends less than 2 seconds to find a solution and the

average relative error is less than "' Furthermore, for more than one third of the test

instances, GA finds solutions better than CPLEX. For example, for the fourth large-sized

instance, GA finds a solution with an objective function value which is about 5.1' lower

than that of the solution found by CPLEX.










Table 4-1. Near-optimal solutions obtained by using CPLEX


Problem size
(D, B)
Instance
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

27
28
29
30
No. of instances
solved by CPLEX


Small
(30, 750)
Objective Time
0.1964
0.6132
0.2574 >
0.2869
0.4828 >
0.4588
0.2817
0.2484
0.1351
0.4558
0.3447
0.4320
0.2872 >
0.4589
0.5883 >
0.2848
0.2751 >
0.1942 >
0.3408 >
0.2600
0.3262 >
0.4457 >
0.1813
0.3991 >
0.6148 >
0.3832 >
0.8989
0.3392
0.1377
0.4448


Medium
(40, 1000)
Objective Time
0.2691
0.4122 >
0.6417 >

0.4141 >
0.7636 >
0.3180 >
0.5691 >
0.4530 >
0.3737 >
0.3347 >
0.4972 >
0.6381 >
0.7216 >
0.6496 >
0.8077 >
0.5183 >
0.2022
0.6919
0.4640 >
0.6155 >
0.5205
0.4363 >
0.1789 >
0.3602 >
0.7231 >
0.7193 >
0.5233 >
0.6959 >
0.4833


Large
(50, 1250)
Objective Time
0.5298 >
1.0745 >
0.5083 >
0.7238 >
1.0353 >
0.2228 >
0.5841 >
0.4073 >
0.6679 >
1.1034 >
0.5721 >
0.6960 >
0.4622 >
0.7431 >
0.4568 >
0.9320 >
0.5609 >
0.5702 >
0.7131 >
0.6928 >
1.0242 >
0.6448 >

0.7533 >
0.7442 >
0. lII 10 >
0.5765 >
0.7700 >
0.4138 >
1.0881 >


(sec.)
37
4
3,600
1,972
3,600
21
1,369
709
2,111
110
1,277
47
3,600
4
3,600
375
3,600
3,600
3,600
8
3,600
3,600
8
3,600
3,600
3,600
12
730
1,770
13


(sec.)
318
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
543
29
3,600
3,600
1,727
3,600
3,600
3,600
3,600
3,600
3,600
3,600
577


(sec.)
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600
3,600


The parameter of population size does not appear to have significant impacts on

relative errors and CPU times. For example, for medium-sized test instances, when the

population size grows from 30 to 90, the average relative error changes from 1.15' to

1.55' and the average CPU time slightly increases from 1.01 seconds to 1.05 seconds.

Moreover, about N I' of the 30 instances obtain the relative errors at the same level with

respect to three different sizes of population.
















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4.5.2 Summary of Results Obtained by GRASP

GRASP is executed for 20D starts (iterations). The threshold parameter, a~, of

construction phase is examined at 0.25, 0.30 and 0.35 three levels. Table 4-3 di pE.l--s

the results of GRASP. As is obvious to see that GRASP takes CPU times longer than

GA to obtain solutions. GRASP spends 3.1 seconds in average to obtain a solution of

the problem with small size, in contrast to GA's 1.6 seconds with respect to the problem

with large size. In general, the accuracy of GRASP and GA does not appear significantly

different, if we compare the best results from each of them. For example, in the problem

with medium size, the relative errors of GRASP (with a~ = 0.25) are 1.;:T' and 8.2 !' in

average and maximum, respectively, and GA (with population size = 40) are 1.15' and

9.45' respectively. Furthermore, among 30 instance GRASP (with a~ = 0.25) finds results

better than CPLEX for 9 instances, compared to 11 instances of GA (with population size

=40).

Table 4-3 presents the tendency that the lower a~ is, the lower the average relative

error is. However, a low value of a~ has the drawback that it may lead the solution to a

local optimal solution and result in a large error. For example, the instance 20 in problem

with medium size has relative error 8.2 !' when a~ is 0.25. In contrast, the instance has

less relative error, 3.'7;' when a~ is increased to 0.35.

In GA and GRASP, we do observe that some instances have relative error greater

than >' such as the instance 14 of the problem with medium size in Table 4-2, and the

instance 20 of the problem with medium size in Table 4-3. To reduce the errors of those

instances, we combines these two algorithms to develop a hybridization.

4.5.3 Summary of Results Obtained by HA

HA is run for [D/10] iterations, in which 200D generations are produced. In the

beginning of each iteration, we generate 2D distinct solutions and pick the best D of these

solutions to form an elite set. The added feature of HA is that a~ decreases as the number

of iterations increases. To evaluate the effects of a~ on the proposed approach, we consider












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three updatingf functions of a~ as follows:


Constant: pl(u) = 0.3,

Linear: p2 (n) = 1.0 0.7 ,and


Nonlinear: p3(u) = 1.0 0.7*

where NV equals [D/10] and is the total number of iterations.

To provide a benchmark, we include the constant function pl(u) in our experiment,

where a~ is set to 0.3 on each iteration a for n = 1,2,..., [D/10]. Using function p2 8), We

linearly decrease a~ to 0.3 as the number of iterations increase. Using function p3 8), Wt

decrease a~ to 0.3 in a non-linear fashion over the iterations. Note that a~ values generated

by function p3 8) 1S Ilo larger than those generated by function p2(u) for each n as shown

in Figure 4-13.

1.0

0.8

0.6 1 ******* pl(n)

0.4 p2(n)

0.2 p3(n)

0.0
1 2 3 4 5



Figure 4-13. The updating functions of a~

The results are shown in Table 4-4. In each test instance, the first column reports the

relative errors and CPU times obtained by using the updating function pl(u), and so on.

For the medium-sized test instances, the average (maximum) relative error decreases from

1 >'.~ (9.3 !'.) to 0.;:".'.~ (4.11 J.'.), when p2 8) is used instead of pl(u) The function p3 8 ,

which has smaller a~ than p2(u) at each iteration n, does not appear to make HA perform

better than the HA with p2 8). FOT eXample, the average relative error for large-sized

instances increases from 0.35'.~ to 0.4!'.~ if the p2 8) is replaced by p3(n). This is different















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from the observation in GRASP, where lower a~ tends yield a lower average relative error.

This also supports the viewpoint mentioned in GRASP, i.e., with a low value of a~, the

approach may converge to a local optimum prematurely.

HA outperforms both GA and GRASP in terms of solution quality. The relative error

is less than 0.5' in average and 5' in maximum for all problems whether p2(u) or p3 8) is

used. Furthermore, HA (with p2(u) or p3(u)) finds better solutions than CPLEX for about

half of the instances, compared to GA or GRASP's one third. As to the computational

requirement, HA spends less than 80 seconds to obtain a solution for the large-sized

instances, which is larger than either GA or GRASP, but is still at an acceptable level

from a practical perspective. From these, we can conclude that HA is an effective and

efficient solution approach for the HBA problem.

4.6 Concluding Remarks and Future Research Directions

In this chapter, we focused on allocating the available .I_ egate hospital bed

capacity among MCUs to balance the bed occupancies among different units. To take

the uncertainties associated with health care systems into account, closed-form results

from queuing theory were incorporated into an optimization framework, which resulted

in large scale nonlinear integer programming formulations for the HBA problem. To

efficiently solve the problem, we proposed three solution approaches. Our computational

study showed that GA and GRASP were very efficient, both of which solved the problem

within 30 seconds with average and maximum errors less than ;:' and 10I' ., respectively.

To decrease the maximum relative errors observed in GA or GRASP, we proposed a

hybridization of GA and GRASP, denoted by HA. HA outperformed either GA or GRASP

in terms of accuracy, whose relative error is <0.5' and <5 in average and in maximum,

respectively, by spending <80 seconds in terms of CPU time. In summary, HA is an

efficient approach that finds good quality solutions for the HBA problem.

In our work, each MCU is modeled as an M/M/~c/c queueing system. Future research

can consider different queueing system configurations with general arrival and service










processes or queueing networks. The design of the queueing network can model the

hierarchical relation between sub-units in each 1\CU, i.e., patients flow through units in

a specific order. We note that the mathematical expressions of stationary probabilities

of a queueing network would be more complicated than that of an Af/Af/c/c queueing

system. Specifically, the expressions of stationary probabilities of a queueing network have

a product form of capacity allocated in each station, which correspond to the the decision

variables of the HBA problem. Another practically relevant research direction is concerned

with the modeling of the blocking behavior between sub-units in each 1\CU. Note that

the blocking in a health care system is different from that in a manufacturing system,

where the work of a blocked part can not he started before entering a designated station.

However, the recovery process of a patient who is blocked from entering the bed in a

downstream unit does not stop. For example, a mother-to-be is typically allocated to a

bed in the labor and delivery unit first, and moved to a bed in postpartum unit to recover

from delivery. It may happen that the mother spends her entire recovery time in the labor

and delivery unit and is discharged from there directly. Because she fully recovers in the

labor and delivery unit before a bed becomes available in the postpartum unit.










CHAPTER 5
EMERGENCY ROOM SERVICES FACILITY LOCATION AND CAPACITY
PLANNING,

5.1 Introduction

This chapter introduces the emergency room services facility location and capacity

planning (ER SFLCP) problem. In traditional facility location models, facility location

and demand allocation decisions are made based on the objectives of minimizing the total

number of facilities opened or minimizing the total or weighted) distance traveled. This

approach ignores the fact that some demand may not he satisfied due to a shortage of

capacity or system congestion as the system operates in real-time.

In a health care service facility, when the emergency room (ER) is full or all intensive

care beds are occupied, hospitals send out divert status. When a hospital is on divert

status, incoming patients might he sent to hospitals which are farther away or kept at

the hospitals where they currently are that may not able to provide adequate service. To

a critical patient, the consequence of divert status can he the difference between life and

death.

The purpose of this work is to construct a facility location model, which simultaneously

determines the number of facilities opened and their respective locations as well as the

capacity levels of the facilities so that the probability that all servers in a facility are busy

does not exceed a pre-determined level. In other words, we want to locate ER services

on a network and determine their respective capacity levels such that the probability of

diverting patients is not larger than a particular threshold.

The remainder of this chapter is organized as follows. In Section 5.2, we review

the related literature. Section 5.3 presents mathematical programming formulation for

ERSFLCP. Section 5.4 details the Lagfrangfian relaxation algorithm that we propose for the

ERSFLCP problem. In Section 5.5, we present results from our computational study that

evaluates the computational performance of the Lagrangian relaxation algorithm. Section

5.6 includes a discussion of the results and -II__- -; future research directions.










5.2 Literature Review

Traditional facility location models (such as the set covering model, the P-median

model, or the P-center model) focus on determining the location of facilities and the

allocation of demand to facilities with the objective of either minimizing the number of

facilities opened, minimizing the sum of fixed facility location and/or transportation costs,

or maximizing the demand covered. In practice, however, a customer may choose to go

to a facility different from the one identified by the optimization model for a number of

reasons, such as the designated facility experiences a temporary shutdown, is associated

with long queues and wait times.

There is a rich body of literature that develop robust facility location models under

uncertainty to hedge the randomness on costs, demands, and travel times utilizing the

robust or stochastic optimization approach. Snyder [103] presents a detailed review of on

stochastic and robust facility location models.

In the area of health care application, Beraldi et al. [12] investigate the problem

of characterizing the optimal locations of emergency medical service sites and numbers

of emergency vehicles required for each site. They consider the problem formulation

in a stochastic optimization setting, where the ability to cover the random requests of

emergency service at demand points is restricted by a set of probabilistic constraints.

Specifically, the probabilistic constraints ensure that the probability that the number

of vehicles located at a facility can cover the random service request is greater than a

prescribed probability value. Some papers consider the hierarchical relation between

facilities in a health care setting. For example, K~oizumi et al. [70] classify mental health

care system into levels of extended acute hospitals, residential facilities and supported

housing, and patients flow through the levels according to their health conditions. Because

of the hierarchical structure of health care service, the capacity requirements of units in a

higher level of hierarchy usually correlate with the units in a lower level. Galviio et al. [39,

40] formulate a hierarchical location-allocation model to determine the capacity required










in perinatal health care facilities, which are categorized into three levels: basic units,

maternity homes and neonatal clinics. They develop a mixed integer linear programming

model which aims to determine the optimal locations of facilities for each level and

allocations of mothers-to-be to these locations for needed type of service.

Another branch of literature that is related to our work develops facility location

models which require backup (multiple) coverage for each demand point. Snyder and

Daskin [104] consider a P-median-based model that minimizes total costs associated with

a location-allocation plan and the expected failure cost. As each demand point is assigned

primary and backup facilities, the expected failure cost is quantified by the additional

transportation cost incurred to cover the demand by the back-up facility. The problem

is formulated as a 0-1 integer programming formulation and solved by the Lagrangian

relaxation algorithm. Jia et al. [56] present a detailed review of traditional facility location

models and propose a general facility location model suited for large-scale emergencies.

In their model, a demand point is considered to be covered if a pre-specified number of

facilities are assigned.

To address the issue of service quality, some papers incorporate queuing systems into

facility location models to consider the randomness on availability of servers and focus on

reducing the demand lost due to the shortage of capacity or system congestion. Marianov

and ReVelle [81] formulate a maximal availability location model, which uses a M/M/~c/c

queueing system to model the server availability of a demand point. The model wants to

locate a set of ambulances such that the demand covered is maximized, where a demand

point is considered to be covered, if the probability that there is an ambulance nearby and

available is greater than a threshold. They show how to transform the nonlinear queueing

expression to an equivalent linear one, and solve the problem by using a commercial solver

(LINDO). Berman et al. [13] model each facility as an M/M/1/la queueing system, where

a is the maximum number of customers allowed in the facility, and consider a facility

location problem with the upper bounds on the amount of demand lost due to insufficient









coverage and system congestion. Berman et al. [14] also investigate the facility location

problem under the objective of maximizing captured demand. They, again, model each

facility as an M/M/1/la queueing system and assume that a customer is lost if the closest

facility and all other facilities that he/or she can reach are full. Both papers ([13, 14])

obtain the solutions, i.e., the location of the facilities, through heuristic approaches.

We note that the capacities of facilities to be opened has an impact on both the total

number and the locations of facilities, particularly when the congestion associated with

the potential facilities is taken into account. In what follows, we model each facility as an

M/M/~c/c queueing system, where c is number of servers in the facility, which designates

the capacity of the facility. The goal of our model is to identify locations of facilities and

specify the capacity levels of the facilities simultaneously.

5.3 Problem Formulation

We consider a network G = (N, A), where N = {1, 2, ... N} is the set of nodes, and

A denotes the set of arcs. We assume that at each node i eN the occurrence of patients

needing ER services is Poisson distributed with rate us, and all patients at a node need to

be directed to one open ER service facility. We consider that an ER service facility can

be opened at any node j E N, and at most p facilities can be opened for providing the

ER services to patients, which is similar to most of the facility location models. Let dij be

the distance (measured in time units) between nodes i and j for i, jE N, and d be the

coverage range of an ER. The allocation parameter aij takes the value of one if dij < d,

zero otherwise. Let t represents the monetary value for travel time for a critical patient.

Let fj be the fixed cost of opening a facility at node j e N, and Cyk, be the operating

cost of the facility j with capacity level k E K, where K = {1, 2,..., K} is the set of

capacity levels of an opened ER. In this problem, the capacity is measured as number of

beds in an ER. That is, if an ER is to be opened at capacity level k E K, then there are

mk beds in the ER. We assume that the lengths of stay of a patient in the ER at node

j are exponentially distributed with rate py, and each ER is modeled as an M/M/~c/c










queueing system. In other words, when all beds in an ER are occupied, patients are

diverted (rejected) to other ERs. Let y be the upper bound of probability of diverting

a patient from an ER, As be the patient arrival rate to an ER service facility at node j,

and zy be the bed capacity of the ER service facility at node j. Note that zy = mk, if the

facility at location j is chosen to be opened at capacity level k.

We have four sets of decision variables. The first set is patient allocation variable xij,

which takes the value of one if node i is served by the ER service facility at location j,

zero otherwise. The second set is ER location and capacity allocation variable yjk, Which

takes the value of one if an ER service facility is placed at node j and operated at capacity

level k, zero otherwise. The last two sets are As and zy, which are determined once the

values of xij and yjk, are aSSigned.

The ERSFLCP problem can be formulated as a nonlinear integer programming

formulation as follows:


min tindijxij + ) ( fj + cyk)#jk (5-1)
i= 1 j= 1 j= 1 k= 1
subject to

ax = 1 Vi(5-2)
j=1

xis < yjk Vi, j(5-3)


yjk < p(5-4)
j=1 k=1

yjk <1 Vj(5-5)



i= 1

Kj(7









K Klj i)(8

k= 1 k= 1
xij, yjk E {0, 1} ,j (5-9)
zj E Z* V (5-10)

where xr(Ay, py, zy) is the probability of diverting patients from an ER service facility at

location j. The formulas for xr(Ay, py, zy) of an M/M/~c/c queueing system can be found

in any queueing book (e.g., Gross and Harris, 1998). Specifically, xr(Ay, py, zy) can be

represented as a function of arrival rate Ay, service rate py, and bed capacity zy, that is

x(A, (,zy =~~ )zy! q! (5-11)11


The objective function (5-1) minimizes the value of time of ER patients and the

costs of opening and operating ER service facilities. Constraint (5-2) stipulates that each

demand node must be covered by one facility. Constraint (5-3) restricts that demand

nodes can only be assigned to opened facilities. Constraint (5-4) imposes the upper bound

on the number of facilities opened. Constraint (5-5) states that an opened facility must be

associated with a single capacity level. Constraints (5-6) and (5-7) obtain the arrival rates

and capacity levels for all the facilities. Constraint (5-8) imposes the upper bound on the

probability of all beds being occupied for an opened facility. Finally, constraints (5-9) and

(5-10) ensure that decision variables are binary and nonnegative integers, respectively.

Note that the factorial terms in equation (5-11) make the ERSFLCP problem

intractable. To overcome this problem, we replace the constraint (5-8) by constraint

(5-12),

As <... ., +My(1- yj) V k(5-12)

where '.. is the largest value which satisfies inequality (5-13) and equation (5-14).


3, mk m q 5 7 (5-13)
Sk* 4
q=0











My = easy(5-14)
i= 1

As a result, we transform the previous nonlinear integer programming formulation to

a linear binary integer programming formulation as follows:


min" ~ingdaymy + ii(j fy + yk)# (5-15)
i=1 j=1 j=1 k=1
subject to

a x, 1 Vi(5-16)
j= 1

xi < CYjkV (5 17)


yjk < p(5-18)
j=1 k=1

yjk, < V (5-19)



i= 1
xij, yjk E {0, 1} ,j (5-21)

5.4 Solution Approach

The ERSFLCP problem can be solved by commercial MILP solvers, however, it takes

time to obtain the optimal solutions of problems with large size or tight constraints, for

example, large values of NV or small values of p. Here, we develop a Lagfrangfian relaxation

(LR) approach to obtain solutions to the ERSFLCP problem.

5.4.1 Lagrangian Relaxation Approach

The general idea of Lagrangian relaxation is to move hard constraints into the

objective and penalize the objective if the constraints are violated. Figure 5-1 presents

the general procedure we use to obtain a solution to the ERSFLCP problem using a

Lagfrangfian relaxation approach. In Figure 5-1, a is the iteration counter. Also, UB* and

LB* denote the incumbent upper and lower bounds on the objective function value of










Lagrangian dual problem, respectively. Similarly, UB" and LB" denote the upper and

lower bounds on the objective function value of Lagrangian dual problem at iteration n,

respectively. u is the counter of number of iterations that UB* does not improved. Last,

nmax and umax are upper bounds of n and u. In the following subsections, we discuss the

elements of the iterative approach.

pro ced ure Lagrangi anRelaxation()
1 n <- 1; a <- 1; UB* <- oo; LB* <- 0;
2 Initialize Lagfrangfian multipliers;
3 while n < nmax and a < umax do
4 Formulate and solve the Lagfrangfian problem to obtain LB";
5 if LB" > LB* then
6 LB* <- LB";
7 end if
8 Obtain UB" using problem specific approach;
9 if UB" < UB* then
10 UB* <- UB";
11 u <- 0;
12 else
13 a<- a+ 1;
14 end if
15 Revise Lagfrangfian multipliers using subgfradient optimization;
16 n<- a+ 1;
17 end while
end LagfrangfianRelaxation.

Figure 5-1. Pseudo-code of the Lagfrangfian relaxation

5.4.2 Lower Bound

To obtain the lower bound of ERSFLCP, we relax the constraint sets (5-17) and

(5-20) with Lagrange multipliers a~ and P, respectively, where a~ and P are matrices with

sizes NVx NV and NVx K, respectively. The relaxation yields the following Lagrangian

problem (ERSFLCP-LR):



i= 1 j= 1k

(fy ~ ~ .I + Cy as -'i j)yjk pjkn~ i (5-22)
j=1 k=1 i j=1 k=1










subject to

asxi = 1 Vi(5-23)
j=1

yjk < p(5-24)
j=1 k=1

yjk <1 Vj(5-25)

xij, yjk E {, 1} ,j (5-26)

Note that the Lagfrangfian problem can be separated into subproblems LX and LY, where

subproblem LX contains variables xij, and the subproblem LY contains variables yjk, aS

follows:


(LX) min am
i= 1 j= 1
subject to

aixij = 1 V
j=1
xi E {0, 1} i


(LY) min ikj
j=1 k=1
subject to


j=1 k=1






where da inn'day + nmy + ne, CE jkji and cjk jtci Cjk n, s 1 sy -i '-~i pj.

Given a set of a~ and p, both subproblems LX and LY are easy to solve. For

subproblem LX, for each i eN the decision variable xij, is set to one if day, < dij

for all j E N. Similarly, subproblem LY can be solved according to each variable yjk S









contribution to the objective function, while for each j eN at most one yjk, can be set to

one for all k E K, and at most p of the yjk S can be set to one for all j EN and k E K.

Note that if yjk = 1, then cjk < 0 must hold.

5.4.3 Upper Bound

In the beginning of the algorithm, we first generate an upper bound by the heuristic

that will introduced later. Also, at each LR iteration, the heuristic is applied to improve

the lower bound solution from the Lagfrangfian subproblem, i.e., subproblems LX and LY,

to become a feasible solution. Figure 5-2 depicts the pseudo-code of the heuristic.

The heuristic starts by resetting the infeasible solution (X, Y) according to the

strategy selected randomly from following:

1. For each i, if xij > 0, then CE= yjk ; n
2. f Eyjk= 0 thenl xsy- = 0 for a~ll i t N.

The first strategy resets facility variables, yjk, based on demand allocation, xij,

determined from subproblem LX. The facilities are ordered according to their patient

arrival rates, i.e., As = Ci nixij, in non-increasing ordern, and then the first p facilities

are set to open at capacity level one, i.e., yjl = 1. The second strategy resets demand

allocation variables, xij, according to solution yjk, form subproblem LY. That is, if a

facility j is not opened, then the demand nodes allocated to facility j are reset to not

covered, i.e., the corresponding variable xij's are reset to 0 and re-allocate them to other

facilities using the next procedure.

Line 2 resets the capacity level k of the opened facilities such that the facilities is

opened at appropriate capacity level k, i.e., Ci nixij < '.... For an opened facility j, if

the demand allocated exceeds its largest capacity, i.e., Ci nixij > ~jK, then an allocated

demand node i is selected randomly and its xij is reset to 0, until Ci nixij < '.. holds.

Next, the while loop in line 3 in Figure 5-2 ensures that all demand nodes are covered

and the number of facilities opened is less than p. The pseudo-code of covering all demand

nodes is presented in Figure 5-3. We randomly select a node a which has not covered by









procedure GenerateFeasibleSolution(X,Y)
1 ResetSolution(X,Y);
2 Reset capacity level;
3 while CiC xij < N or Cj Ck yjk > p do
4 if CiC xij < NV then
5 CoverDemand(X,Y);
6; end if
7 if Cj Ck yjk > p then
8 CloseFacility(X ,Y);
9 end if
10 end while
11 return (X,Y);
end GenerateFeasibleSolution.

Figure 5-2. Pseudo-code of the feasible solution generation

any facility, and generate a set OpenF which contains the facilities meeting the following

criteria:

1. The facility j is opened, i.e., Ck yjk, > 0;
2. The facility j can covered the demand node n, i.e., any = 1; and
3. After allocating node a to the facility j, the sum of arrival rates of the allocated
demand node does not exceed facility j's maximum capacity, i.e., Ci nixij +n, < < jK

If the set OpenF is empty, then a set NVotOpenF is generated. The set NVotOpenF

includes the facilities which are not opened, i.e., Ck yjk = 0, and the node u is within

their coverage range, i.e., any = 1 for all j E N. Then, in the line 8 in Figure 5-3 we open

a facility je N otOpenF to cover node u. There are many strategies that we can apply

to choose which facility to open, such as, open the facility je N otOpenF which is the

closest one to node n, the one with the largest capacity, or the one with the smallest fix

cost. Here we apply the three strategies and find the corresponding facilities for each of

them. If these strategies yield different facility options, we randomly select one of them

to open. Once all demand nodes are covered by the opened facilities, the capacity level of

each opened facility is reset to appropriate level k E K (line 14 in Figure 5-3) such that



So far, we have ensured that all demand nodes are covered by facilities which are

opened, and the demand allocation does not exceed the maximum capacity of the opened









procedure CoverDemand(X,Y)
1 while Ci C xi < NV do
2 a <- RandomlySelectElement (UncoveredNode);
3 Generate set OpenedF = {j : Ck yjk > 0, any = 1, Ci nixij + n, < < jK~j E 6
4 if OpenF / 0 then
5 j <- the closest facility j E OpenF;
6i else
7 Generate NVotOpenedF = {j : Ck yjk = 0, any = 1, jE N};
8 select j from the set NVotOpenF;
9 NVotOpenF <- NVotOpenF \ j;
10 OpenF <- NVotOpenF U j;
11 end if
12 xij <- 1;
13 end while
14 Reset capacity level;
15 return (X,Y);
end CoverDemand;

Figure 5-3. Pseudo-code of covering all demand nodes

facilities. The next thing to do is to inspect whether the number of facilities opened is no

larger than p. If the number of facilities opened is less than or equal to p, then a feasible

solution is generated. Otherwise, one of the opened facilities is selected randomly and

closed, until the number of facilities opened is no larger than p. In addition, the associated

demand nodes are reset to not covered, i.e., reset xij = 0, where facility j is chosen to

close.

procedure CloseFacility(X,Y)
1Generate set OpenedF = {j : Ck yjk > 0, jE N};
2 while Cj Ck yjk > p do
3 j <- RandomlySelectElement (OpenF);
4 yjk <- 0, for all k E K;
5 for i = 1 to NV do
6 xij <- 0;
7 end for
8 OpenF <- OpenF \ j;
9 end while
10 Reset X based on Y;
11 return (X,Y);
end CloseFacility;

Figure 5-4. Pseudo-code of closing facilities










The procedures of CoverDemand and CloseFacility are repeated, until a feasible

solution is generated.

5.4.4 Lagrangian Multipliers

Due to the modern software and hardware developments, the linear programming

problems can be efficiently solved by any commercial LP solvers. To take advantage

of the existing tools, we use the dual information of the linear form of the ERSFLCP

problem, denoted by ERSFLCPr, to initialize the Lagrangian multipliers, rather than

starting the algorithms from scratch as traditional approaches. The ERSFLCP, problem

is generated by replacing the constraint sets (5-17) and (5-20) by 0 < xij < 1 and

0 < yjk I i, TOSpectively. Then, CPLEX is used to solve the ERSFLCP, problem, and the

dual information of the constraint sets 0 < xij < 1 and 0 < yjk I 1 are eXtracted to set the

initial values of Lagfrangfian multipliers. Let or, and P, be the dual values associated with

the constraint sets 0 < xij < 1 and 0 < yjk I i, TOSpectively. The multipliers t~o and Po of

ERSFLCP-LR are initialized by the equations t~o = -ar, and Po -P,

We then apply the method described by Fisher (1981) to update the multipliers. At

each iteration n, the step size t" is obtained by

B",- "
S=


where e1 < B" < 2 e2 6~1, 62 0 ), 8 is the best upper bound of the optimal objective

value of ERSFLCP-LR found, and 0" is the the optimal objective value of ERSFLCP-LR

at iteration n. Then, the Lagrangian multipliers are reset by





nn"+1pnt = pjk" + s"(nixij .. >;.l, My (1 yjk))Vk

The iterations are stopped when the number of iteration exceeds a prespecified value,

n,,,, or the upper bound is not improved for umax iterations.









5.5 Computational Study

In this section, we present results from our computational study, where we investigate

the computational performance of the Lagrangian relaxation approach on the ERSFLCP

problem. The computational experiments are implemented on a workstation with two

Pentium 4 3.2 GHz processor and 6 GB of memory.

5.5.1 Experimental Design

In our study, we conduct experiments on four factors including the maximum number

of facilities opened (p), number of capacity levels (K), the incremental amount of beds

associated with each capacity level (A), probability of diverting a patient of an opened

facility (y), and the monetary value for travel time for a patient (t). For each experiment,

we consider three networks consisting of 25, 50 and 100 nodes, respectively, Furthermore,

one of the factor is tested at three levels as listed in Table 5-1, and the other three

parameters are set at level 2.

Table 5-1. Experimental factor settings
Parameters Level 1 Level 2 Level 3
Experiment 1 p 5 10 20
Experiment 2 K( (A) 5 (8) 8 (5) 10 (4)
Experiment 3 y 0.5' 1 2
Experiment 4 t ($/ per minute) 25 50 100

For each experimental setting, we generate 30 random instances. Each random

instance is obtained by generating parameters that correspond to patient arrival rates,

service rates, fixed and operating costs and distance between each node pair. Specifically,

each parameter is obtained by using the formula: u(mean value of the parameter), where u

is drawn from the distribution U [0.5, 1.5]. The mean values of the random parameters are

listed as follows:

* Mean arrival rate (us): 1.5 persons/per hour for each demand node i;
* Mean service rate (py): 0.5 persons/per hour for the facility opened at location j;
* Mean fixed cost (fj): $5000 /per d~i- for the facility opened at location j;
* Mean operating cost per bed: $1000 /per d~i- for the facility opened at location j;
and










*Mean distance between each node pair (dij): 50 minutes.

Last, the coverage range, d, is set to 50 minutes for all instances.

For each instance, the Lagrangian relaxation algorithm described in Section 5.4 is

applied. The upper limit of the number of LR iterations (nm=,) is set at 100,000 and

the upper limit of the number of consecutive iterations fail to improve the best known

feasible solution (um=,) is set at 10,000. The parameter B used to modified the step size is

initialized at 2, and divided by 1.5 if the lower bound is not improved for 3NV iterations.

5.5.2 Experimental Results

Before presenting the experimental results, we use one of the instances in experiments

to show the benefit of utilizing the dual information of ERSFLCP, in LR. The solid lines

in Figure 5-5 depict the upper and lower bounds, respectively, obtained from the LR

iterations with the dual information of ERSFLCPr. The dash lines show the LR results

without applying the dual information of ERSFLCPr. The dual information of ERSFLCP,

provides a good lower bound solution which guides the heuristic to find the best upper

bound solution earlier than the guidance of the lower bound obtained from using the

Lagfrangfian multipliers generated from scratch. The convergence gaps in Figure 5-5 are

given by

x 100,

where 8 and 8" are the upper and lower bounds of optimal objective value of ERSFLCP-LR

obtained from the LR algorithm.

Table 5-2 to Table 5-5 report the results of four experiments. In general, LR performs

quite well in solving the ERSFLCP problem in terms of the convergence gap and the

computational effort (i.e., CPU time) required. For the network with 25 nodes, LR takes

less 2 seconds to obtain the solutions with convergence gaps less than 5' on average. For

the largest network tested, i.e., 100 nodes, the average CPU time is increased, although

it is still less than 60 seconds. Moreover, LR does not appear to experience significant

increase in convergence gap due to an increase in problem size.














I ~I -UB (dual)

Gap =4% Gap =15%
L~4()(()(()( .. --LB (dual)


~a" ~~UB (scratch)
O 2()(()(()(

----LB (scratch)

() 5()()( 1()(()( 15()()( 2()(()( 25()()( 3()(()(
Iteration

Figure 5-5. Convergence speed of the modified LR
The first experiment shows that the smaller the value of p is, the higher the CPIT

time required to obtain the solution, since smaller p results in a tighter constraint on

number of facilities opened. In particular, this trend can he observed in the problems with

small- and niedium-size networks. Nevertheless, the problem can still be solved within 40

seconds even for large-size networks.

Table 5-2. Experiment 1: effects of nmaxiniun number of facilities opened
Network size 25 50 100 Average
Gap Time Gap Time Gap Time Gap Time

5 :3.0 1.1 4.4 4.9 5.8 41.8 4.4 15.9
10 :3.1 0.9 :3.0 4.3 :3.8 :35.5 :3.3 1:3.6
20 2.5 0.9 :3.3 :3.2 2.9 29.4 2.9 11.2
Average 2.9 1.0 :3.6; 4.1 4.2 :35.6 :3.5 1:3.6


In the second experiment, we assume that at most 40 heds are used for an opened

ER, so the incremental amount of beds per capacity level are changed with the setting

of K. A large K value represents that capacity is increased at small scale, i.e., small

batch size, and increases the number of variables of the underlying formulation the

the ERSFLCP problem. As a result, the CPIT time is increased as K increases, while










the growth rate of CPU time is less than the growth rate of K. The impact of K on

convergence gap is not significant, especially, with respect to the problem with large size.

Table 5-3. Experiment 2: effects of capacity setting
Network size 25 50 100 Average
Gap Time Gap Time Gap Time Gap Time
K ( (sec.) (sec.) (sec.) (sec.)
5 (8) 0.7 0.8 2.2 4.2 3.6 31.4 2.1 12.1
8 (5) 3.2 1.1 2.9 4.1 3.9 34.0 3.3 13.1
10 (4) 3.6 1.1 3.3 4.5 3.9 35.9 3.6 13.8
Average 2.5 1.0 2.8 4.3 3.8 33.8 3.0 13.0

The third experiment investigates the impact of the diversion probability on the

performance of the proposed LR approach. Given the capacity level k, the smaller y incurs

the smaller value of the maximum arrival rates that a facility is able to serve, i.e., '., in

constraint set (5-20). That is, the constraint set (5-20) becomes tighter as y decreases.

As shown in Table 5-4, the CPU time slightly increases as y decreases from 2.01' to

1.'.However, when the value of y is further reduced to 0.5' the required CPU time

decreases, which is different from the trend that we observe before. The convergence gap

does not appear to be affected by y significantly, in average, the gaps are 3.10' and 2.91' .

when y's are 0.5' and 2.01' respectivley.

Table 5-4. Experiment 3: effects of diversion probability
Network size 25 50 100 Average
Gap Time Gap Time Gap Time Gap Time
Y '. (sec.) (sec.) (sec.) (sec.)
0.5 3.7 0.9 3.3 4.1 3.8 34.9 3.6 13.3
1.0 3.1 1.0 3.0 4.3 3.9 56.6 3.3 20.6
2.0 2.4 0.9 2.7 3.6 3.5 48.3 2.9 17.6
Average 3.1 0.9 3.0 4.0 3.7 46.6 3.3 17.2

The last experiment explores the effects of patients' value of time t on the performance

of the proposed LR approach. Table 5-5 shows that the higher the patients' time value is,

the smaller the convergence gap is. We also observe that the CPU time is not significantly

affected by the change of t.









Table 5-5. Experiment 4: effects of time value
Network size 25 50 100 Average
Gap Time Gap Time Gap Time Gap Time
t ( ) (sec.) (sec.) (sec.) (sec.)
25 4.9 1.1 4.9 5.8 5.6 45.8 5.1 17.6
50 3.0 1.1 2.9 4.7 3.8 49.0 3.2 18.2
100 1.8 1.1 1.7 5.4 2.7 50.8 2.1 19.1
Average 3.2 1.1 3.2 5.3 4.0 48.5 3.5 18.3

To illustrate the performance of the heuristic developed for obtaining the upper

bound (feasible) solution at each iteration, we use CPLEX to solve the ERSFLCP

problem. The parameters, p, K, y and t are set to their respective values for level 2 in

Table 5-1. For each test instance, CPLEX is stopped if the relative stopping tolerance of

0.01 is satisfied, or CPU time of 3,600 seconds is used. Table 5-6 summarizes the results,

where the relative error of each instance is given by

Obj of CPLEX Upper bound of LR
x 100I .
Obj of CPLEX

Table 5-6 shows that the developed heuristic is very effective. For small- and median-size

problems, the upper bounds obtained from the heuristic are very close to the optimal

objective values, the average and maximum relative errors are less than 0.;:' and 1.5' .

respectively. For the problem with largest size, CPLEX is not able to find the optimal

solution within 3,600 seconds. Therefore, we use the best results that CPLEX can find

within 3,600 seconds to compare with the results obtained by heuristic. The comparison

further ensures the effectiveness of the heuristic developed here. The heuristic not only

spends much less time (36 seconds in average) than CPLEX, but also find solutions with

better quality in 50I' of the instances.

5.6 Concluding Remarks and Future Research Directions

In this chapter, we developed a model for the emergency room services facility

location and capacity planning (ERSFLCP) problem, in which each facility is modeled

as an M/M/~c/c queueing system to consider the impact of the uncertainties associated

with patient arrivals and lengths of stay. The model was designed to simultaneously


















Table 5-6. Performance of the heuristic


Nodes

Instance
1
2
3
4
5

7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Min.
Avg.
Max.


25
CPLEX Heuristic Err.


50
CPLEX Heuristic Err.


100
CPLEX Heuristic Err.


(sec.)
0.4
0.7
0.3
0.5
0.8
0.7
0.2
0.2
0.9
0.2
0.8
0.3
0.6
0.3

0.5
0.6
0.8
0.7
0.8
0.4
0.4
0.2
0.3
0.5
0.8
0.6

0.2
0.1
0.1
0.5
0.9


(sec.)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1


(sec.)
31.8
84.3
44.9
109.9
82.4
47.9
115.0
18.7
27.7
32.6
130.9
65.2
65.5
39.9
25.6
456.4
47.9
34.7
47.3
119.8
53.5
46.7
17.4
62.8
502.1
53.6
27.4
146.6
39.9
176.9
17.4
91.8
502.1


(sec.)
3.3
4.5
3.5
4.7
5.1
7.5
3.5
4.1
3.5
4.6
5.1
5.3
3.0
4.8
3.4
3.7
4.3
3.2
3.5
3.8
3.8
3.4
7.5
3.3
3.5
3.6
3.7
4.8
5.8
4.1
3.0
4.3
7.5


(sec.)
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600
>3600


(sec.)
26.6
59.7
26.0
39.7
31.9
33.3
29.4
30.5
51.4
27.8
65.6
30.1
22.8
39.5
25.9
55.3
34.8
41.5
19.5
24.4
25.7
25.4
33.7
49.6
28.1
23.6
19.9
37.1
60.2
46.2
19.5
35.5
65.6










determine the number of facilities opened and their respective locations as well as the

capacity levels of the facilities so that the probability of diverting patients is not larger

than a particular threshold. A Lagrangian relaxation approach was proposed to obtain

solutions of the ERSFLCP problem. The relaxation scheme proposed yield a separable

Lagrangian problem that is easy to solve. The upper bound at each LR iteration was

obtained by a heuristic developed in this work. In addition, to speed up the convergence

process, we -II- -- -1h I1 to use the dual information of the linear progranining relaxation

of the ER SFLCP problem to generate the initial set of Lagrangian multipliers. The

computational study demonstrated that the ERSFLCP problem can he efficiently solved

by the Lagrangian relaxation algorithm, and the developed heuristic provides upper bound

solutions with good quality.

An ininediate extension of our model is to include the closest-assignnient constraints

which ensure that each demand point is allocated to the closest open facility. Some other

practically relevant variations are concerned with the alternative objective functions, such

as profit nmaxintization, and travel and service times nxinintization. Another closely related

problem is that given some ER service facilities are opened already and B new heds are

considered to be added at existing facilities or newly opened facilities. We note that this

can he defined by adding the constraints on capacity available, distinguishing the set of

existing and new ER locations, and replacing the fixed cost to capacity expansion cost of

existing facilities.









CHAPTER 6
CONCLUSIONS AND FITTIRE RESEARCH DIRECTIONS

In this work, we have presented the integrated use of optimization and queueing

theory to determine the optimal capacity plan for health care systems.

C'!s Ilter 2 detailed the aggregate hospital bed *r'i'- .:; -111.1.rt:e..t (AHBCP) problem

and a network flow approach to specify the optimal bed capacity planning decisions

throughout a finite planning horizon for hospitals. In this chapter, a hospital was modeled

as a G/G/c queueing system with a single bed type and a single patient class. We

demonstrated that for realistic-sized capacity planning problems, our network formulation

is not computationally intensive, and allows us to obtain optimal bed capacity plans

quickly.

C'!s Ilter 3 introduced the health care team -r'/'- I 1'1.r t. e.. t. (HC TCP) problem, in

which the underlying queueing system was more complex than the one used for AHBCP.

In particular, we considered a queueing system where there are two classes of patients

and two types of care teams, where service rates are patient-class dependent and one

type of care team can substitute for the other. We developed queueing models for both

preemptive and non-preemptive cases, and developed approximation procedures to

estimate the average time that each patient class spends in the system. The results from

approximation method were then incorporated into the optimization model to determine

the minimal cost capacity plan of health care teams throughout a finite planning horizon.

Our computational study showed that our approximation approach provides sufficiently

accurate results that can he used in practice to make long-term health care team service

capacity planning decisions.

After the .I_ egate bed capacity was specified in C'!s Ilter 2, we developed the

hospital bed allocation (HBA) model to obtain the balanced bed allocation among

different medical care units, which were modeled as M/M/~c/c queueing systems.

To efficiently solve the problem, we proposed three solution approaches including










genetic algorithm(GA), greedy randomized adaptive search procedure (GRASP) and

a hybridization of GA and GR ASP (HA). Our computational study showed that the

proposed algorithms can solve the problem within a short time while providing solutions

with high quality for large-sized, realistic test instances.

('!, Ilter 5 developed a emer i. ,...;; room services f r. .:1.:1 location and -r'i'- I rk'*-'":':: T

(ERSFLCP) model, where each emergency room service facility is viewed as a Af/Af/c/c

queueing system. The model was designed to simultaneously determine the number of

facilities opened and their respective locations as well as the capacity levels of the facilities

(capture in terms of number of beds) so that the probability that the facility is full and

incoming patients have to be diverted (i.e., diversion probability) is not larger than a

particular threshold. A Lagrangian relaxation approach was proposed to obtain facility

locations and capacity plan. The experimental results illustrated that the Lagrangian

relaxation algorithm is very efficient in solving the problem, and the developed heuristic

provides solutions with good quality.

In our work to date, we primarily focused on strategic level, long-term planning

decisions where we made some simplifying assumptions as to the capabilities of the

resources and the arrival rates and lengths of stay for the patients. Specifically, we

assumed that the available beds in a service (in ('! .pters 3, 4, and 5) or the hospital (in

('!s Ilter 2) are identical. Similarly, we assumed that the arrival rates and length of stay

for the patients are identical (in C'!s Ilters 2 and 5). But we also considered the case where

patients can he grouped into two classes (each of which corresponds to an acuity level) or

into multiple classes (each of which corresponds to a particular speciality) to model arrival

rates and lengths of stay. As we mainly focused on strategic level decision 1!! I1:;0s these

assumptions are justifiable. However, for more detailed planning there is a need to take

other realistic considerations into account.

There are typically multiple types of beds (e.g., adult intensive care beds, pediatric

intensive care beds, burn intensive care beds, medical/surgical beds), i.e., multiple










units, in a service or a hospital. Typically, different units are used to accommodate the

patients during different phases of the treatment. Therefore, for more detailed capacity

analysis, there is a need to distinguish between these units and model how the patients

flow through these units. Another important concern in this context is the consideration

of interaction between different units as well as services. That is, if there is no enough

capacity in a speciality service, the patient can he accommodated in another speciality

service. This, in turn, increases the effective service rate of the speciality and the traffic of

another speciality. Similarly, if there is not enough capacity in the downstream unit within

a service, then the patient can continue treatment in the upstream unit, i.e., blocking

Therefore, for more detailed capacity planning, there is a need to consider multiple types

of resources, multiple types of patients, and multiple modes of interaction between units

and services.

In our work, we have primarily focused on the objective of minimizing total costs (in

('!s Ilters 2, 3, and 5) and balancing work load across units (in ('! .pter 4). Nowced a7

hospitals are focusing on revenue management practices to improve their financial

situation. This modern revenue management culture requires hospital administrators

to focus on maximizing profit rather than on minimizing operating costs. Therefore,

future work should examine the revenue aspects of hospital operations and focus on profit

maximization type objectives. However, this shift in emphasis from cost minimization to

profit maximization should not ignore the quality aspects associated with the delivery of

health care services. In our research, we focused on timeliness (e.g., average service time,

average waiting time) and access (e.g., diversion probability) to quantify service quality.

Future research in the area should concentrate on developing other metrics to model other

aspects of service quality, such as patient safety and service effectiveness.









REFERENCES

[1] Agency for Healthcare Research and Quality. Last time accessed: November 2007,
http://www.ahrq.gov/research/havbed/defintoshm

[2] R.K(. Al.0l I T.L. Magnanti, J.B. Orlin, Network Flows: Theory, Algorithms, and
Applications, Prentice Hall, New Jersey, 1993.

[3] L.A. Aiken, S.P. Clarke, D.M. Sloane et al., Nurses reports on hospital care in five
countries, Health Affairs 20 (2001) 43-53.

[4] L.A. Aiken, S.P. Clarke, D.M. Sloane et al., Hospital nurse staffing and patient
mortality, nurse burnout, and job dissatisfaction, JAMA 288 (2002) 1987-1993.

[5] E. Akgahl, M.J. C~tid, C.-I Lin, A network flow approach to optimizing hospital bed
capacity decisions, Health Care Management Science 9 (2006) 391-404.

[6] R. Akkerman, M. K~nip, Reallocation of beds to reduce waiting time for cardiac
surgery, Health Care Management Science 7 (2004) 119-126.

[7] American Hospital Association, Hospital Statistics 2005 Edition, Health Forum,
Chicago, 2005.

[8] P.-A. Andersson, E. Varde, F. Diderichsen, Modelling of resource allocation to health
care authorities in Stockholm County, Health Care Management Science 3 (2000)
141-149.

[9] R. Batta, O. Berman, Q. W-,v._ Staffing and switching costs in a service center with
flexible servers, European Journal of Operational Research 177 (2007) 924-938.

[10] G.J. Bazzoli, L.R. Brewster, G. Liu, S. K~uo, Does U.S. hospital capacity need to be
expanded? Health Affairs 22 (2003) 40-54.

[11] D. Bellandi, Running at capacity, Modern Healthcare 110 (1999) 112-113.

[12] P. Beraldi,M.E. Bruni, D. Conforti, Designing robust emergency medical service
via stochastic programming, European Journal of Operational Research 158 (2004)
183-193.

[13] O. Berman, D. K~rass, J. Wang, Locating Service Facilities to Reduce Loss Demand,
IIE Transactions 38 (2006) 933-946.

[14] O. Berman, R. Huang, S. K~im, S., M. Menezes, Locating capacitated facilities to
maximize captured demand, IIE Transactions 39 (2007) 1015-1029.

[15] D.M. Berwick, We can cut costs and improve care at the same time, Medical
Economics 180 (1996) 185-187.

[16] G.R. Bitran, D. Tirupati, Capacity planning in manufacturing networks with
discrete options, Annals of Operations Research 17 (1989) 119-136.









[17] G.R. Bitran, D. Tirupati, Tradeoff curves, targeting and balancing in manufacturing
queueing networks, Operations Research 37 (1989) 547-564.

[18] J.T. Blake, M.W. Carter, A goal programming approach to strategic resource
allocation in acute care hospitals, European Journal of Operational Research 140
(2002) 541-561.

[19] J.P.C. Blanc, Performance evaluation of polling systems by means of the power-series
algorithm, Annals of Operations Research 35 (1992) 155-186.

[20] J.P.C. Blanc, Performance analysis and optimization with the power-series
algorithm, in L. Donatiello, R. Nelson (eds), Performance analysis and optimization
with power series algorithm Springer \: 11 I_ Berlin, 53-80, 1993.

[21] B. Boffey, D. Yates, R.D. Galviio, An algorithm to locate perinatal facilities in the
municipality of Rio de Janeiro, Journal of the Operational Research Society 54
(2003) 21-31.

[22] J. Bowers, B. Li-n ix- G. Mould, T. Symonds, Modelling outpatient capacity for a
diagnosis and treatment centre, Health Care Management Science 8 (2005) 205-211.

[23] J. Bowers, G. Mould, Managing uncertainty in orthopaedic trauma theatres,
European Journal of Operational Research 154 (2004) 599-608.

[24] J. Bowers, G. Mould, Ambulatory care and orthopaedic capacity pI l ....ilr_ Health
Care Management Science 8 (2005) 41-47.

[25] M.L. Brandeau, F. Sainfort, W.P. Pierskalla, Operations Research and Health Care:
A Handbook of Methods and Applications, Springer, Boston, 2004.

[26] K(.M. Bretthauer, M.J. Ci~ti, A model for planning resource requirements in health
care organizations, Decision Sciences 29 (1998) 243-270.

[27] M.J. Brusco, M.J. Showalter, Constrained nurse staff 2.1, l1i--;- OMEGA:
International Journal of Management Science 21 (1993) 175-186.

[28] E.K(. Burke, P. De Causmaecker, G. Vanden Berghe, H. Van Landeghem, The state
of the art of nurse 10 ~--ha li, Journal of Scheduling 7 (2004) 441-499.

[29] F. Cerne, J. Montague, Capacity crisis, Hospitals and Health Networks 68 (19)
(1994) 30-36.

[30] J.K(. Cochran, A. Bharti, Stochastic bed balancing of an obstetrics hospital, Health
Care Management Science 9 (2006) 31-45.

[31] R.C. Coile Jr., Futurescan 2002: a forecast of healthcare trends 20022006, Health
Administration, Chicago, 2002.

[32] M.J. Ci~ti, S.L. Tucker, Four methodologies to improve healthcare demand
forecasting, Healthcare Financial Management 55 (2001) 54-58.









[:33] R. Davies, Simulation for planning services for patients with coronary artery disease,
European Journal of Operational Research 72 (1994) :32:3-332.

[:34] K(. Davis, S.C. Schoenhaunt, A.-M. Audet, A 2020 Vision of Patient-Centered
Primary Care, Journal of General Internal Medicine 20 (2005) 95:3957.

[:35] L. Delesie, A. K~astelein, F. Merode, J.M.H. Vissers, Managing health care under
resource constraints, European Journal of Operational Research 105 (1998) 247-247.

[:36] T.A. Feo, M.G.C. Resende, A probabilistic heuristic for a computationally difficult
set covering problem, Operations Research Letters 8 (1989) 67-71.

[:37] M.L. Fisher, The Lagrangian relaxation method for solving integer progranining
problems, Alanagenient Science 27 (1981) 1-18.

[:38] S. Flessa, Where efficiency saves lives: A linear progranine for the optimal allocation
of health care resources in developing countries, Health Care Alanagenient Science :3
(2000) 249-267.

[:39] R.D. Galvao, L.G.A. Espejo, B. Boffey, A hierarchical model for the location of
perinatal facilities in the municipality of Rio de Janeiro, European Journal of
Operational Research 1:38 (2002) 495-517.

[40] R.D. Galviio, L.G.A. Espejo, B. Boffey, D. Yates, Load balancing and capacity
constraints in a hierarchical location model, European Journal of Operational
Research 172 (2006) 6:31-646.

[41] D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning,
Addison-Wesley, Wokingham, England, 1989.

[42] S.M. Goldstein, P.T. Ward, G.K(. Leong, T.W. Butler, The effect of location,
strategy, and operations technology on hospital performance, Journal of Operations
Alanagenient 20 (2002) 6:375.

[4:3] F. Gorunescu, S.I. McClean, P.H. Millard, Using a queueing model to help plan bed
allocation in a department of geriatric medicine, Health Care Alanagenient Science 5
(2002) :307-312.

[44] L. Green, A queuing system with general-use and linlited-use servers, Operations
Research :33 (1985) 168-182.

[45] L.V. Green, How many hospital beds? Inquiry :38 (2002/200:3) 400-412.

[46] L.V. Green, V. Nguyen, Strategies for cutting hospital beds: the impact on patient
service, Health Services Research :36 (2001) 421-442.

[47] S. Groothuis, A. Hasnian, P.E.J. van Pol et al., Predicting capacities required in
cardiology units for heart failure patients via simulation, Computer methods and
Programs in Biontedicine 74 (2) (2004) 129-141.










[48] D. Gross, C.M. Harris, Fundamentals of Queueing Theory (3nd ed.), John Wiley and
Sons, New York, 1998.

[49] K(. Grumbach, T. Bodenheimer, Can health care teams improve primary care
practice? JAMA: the journal of the American Medical Association 291 (2004)
1246-1251.

[50] C. Hayward, What are we going to do with all our excess capacity? Health Care
Strategic Management 16 (1998) 20-23.

[51] P.R. Harper, A framework for operational modelling of hospital resources, Health
Care Management Science 5 (2002) 165-173.

[52] G.W. Harrison, A. Shafer, M. Mackay, Modelling Variability in Hospital Bed
Occupancy, Health Care Management Science 8 (2005) 325-334.

[53] J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan
Press, Ann Arbor, 1975.

[54] G. Hooghiemstra, M. K~eane, S.V.D. Ree, Power series for stationary distributions
of coupled processor models, SIAM Journal on Applied Mathematics 48 (1988)
1159-1166.

[55] Institute of Medicine, Crossing the Quality ChI I-ms! National A< II1. ...vJ Press,
Washington, D.C, 2001.

[56] H. Jia, F. Ordonez, M.M. Dessouky, A modeling framework for facility location of
medical services for large-scale emergencies, IIE Transactions 39 (2007) 41-55.

[57] A.M. Johnson, Capacity planning for the future, Journal of Health Care Finance 24
(1997) 72-75.

[58] J.B. Jun, S.H. Jacobson, J.R. Swisher, Application of discrete-event simulation in
health care clinics: A survey, Journal of the Operational Research Society 50 (2)
(1999) 109-123.

[59] E.P.C. K~ao, C. Lin, The M/M/1 queue with randomly varying arrival and service
rates: a phase substitution solution, Management Science 35 (1989) 561-570.

[60] E.P.C. K~ao, K(.S. Narayanan, Computing steady-state probabilities of a
nonpreemptive priority multiserver queue, ORSA Journal on Computing 2 (1990)
211-218.

[61] E.P.C. K~ao, M. Queyranne, Budgeting costs of nursing in a hospital, Management
Science 31 (1985) 608-621.

[62] E.P.C. K~ao, G.G. Tung, Aggregate nursing requirement planning in a public health
care delivery system, Socio-Economics Planning Science 15 (1981) 119-127.









[63] E.P.C. K~ao, G.G. Tung, Bed allocation in a public health care delivery system,
Management Science 27 (1981) 507-520.

[64] E.P.C. K~ao, S.D. Wilson, Analysis of nonpreemptive priority queues with multiple
servers and two priority classes, European Journal of Operational Research 118
(1999) 181-193.

[65] E.H. K~aplan, M. Johri, Treatment on demand: an operational model, Health Care
Management Science 3 (2000) 171-183.

[66] S. K~avanagh, J. Cowan, Reducing risk in healthcare teams: an overview, Clinical
Governance 9 (2004) 200-204.

[67] S. K~im, I. Horowitz, K(.K. Young, T.A. Buckley, Analysis of capacity management of
the intensive care unit in a hospital, European Journal of Operational Research 115
(1999) 36-46.

[68] S. K~im, I. Horowitz, K(.K. Young, T.A. Buckley, Flexible bed allocation and
performance in the intensive care unit, Journal of Operations Management 18
(2000) 427-443.

[69] L. K~leinrock, Queueing systems: Theory Vol. 1, Wiley, New York, 1975.

[70] N. K~oizumi, E. K~uno, T.E. Smith, Modeling patient flows using a queuing network
with blocking, Health Care Management Science 8 (2005) 49-60.

[71] G. K~oole, On the use of the power series algorithm for general Markov processes,
with an application to a Petri net, INFORMS Journal on Computing 9 (1997) 51-56.

[72] D.H. K~ropp, R.C. Carlson, Recursive modeling of ambulatory health care settings,
Journal of Medical Systems 1 (1977) 123-135.

[73] R.J. K~usters, P.M.A. Groot, Modelling resource availability in general hospitals
design and implementation of a decision support model, European Journal of
Operational Research 88 (1996) 428-445.

[74] G. Latouche, V. Ramaswami, A logarithmic reduction algorithm for
quasi-birth-and-death processes, Journal of Applied Probability 30 (1993) 650-674.

[75] G. Latouche, V. Ramaswami, Introduction to Matrix Analytic Methods, SIAM,
Philadelphia, PA, 1999.

[76] L.X. Li, W.C. Benton, Performance measurement criteria in health care
organizations: Review and future research directions, European Journal of
Operational Research 93 (1996) 449-468.

[77] L.X. Li, W.C. Benton, G.K(. L ..nr The impact of strategic operations management
decisions on community hospital performance, Journal of Operations Management 20
(2002) 389-408.










[78] E. Litvak, P.I. Buerhaus, F. Davidoff, F. et al., Managing unnecessary variability
in patient demand to reduce nursing stress and improve patient safety, Joint
Coninission Journal on Quality and Patient Safety :31 (2005) :330-:338.

[79] 31. Mackay, Practical experience with bed occupancy nianagenient and planning
systems: an Australian view, Health Care Alanagenient Science 4 (2001) 47-56.

[80] 31. Mackay, 31. Lee, ClI u! e. of Models for the Analysis and Forecasting of Hospital
Beds, Health Care Alanagenient Science 8 (2005) 221-230.

[81] V. Alarianov, C. ReVelle, The queueing nmaxinmal availability location problem:
A model for the siting of emergency vehicles, European Journal of Operational
Research 9:3 (1996) 110-120.

[82] A.H. Marshall, S.I. McClean, C.M. Shapcott, P.H. Millard, Modelling patient
duration of stay to facilitate resource nianagenient of geriatric hospitals, Health Care
Alanagenient Science 5 (2002) :31:3-319.

[8:3] A.H. Marshall, C. Vasilakis, E. El-Darzi, Length of -r li--hased patient flow models:
recent developments and future directions, Health Care Alanagenient Science 8
(2005) 21:3220.

[84] B.J. Masterson, T.G. Mihara, G. Miller, S.C. Randolph, M.E. Forkner, A.L. Crouter,
Using models and data to support optimization of the military health system: A
case study in an intensive care unit, Health Care Alanagenient Science 7 (2004)
217-224.

[85] W.E. McAleer, I.A. Naqvi, The relocation of ambulance stations: A successful case
study, European Journal of Operational Research 75 (1994) 582-588.

[86] L.W. Morton, N. Bills, D. K~ay, Boosting local economies
through health care economic development, Last time
accessed: August 2005, http://www.cardi. cornell.edu/
economic c_development/community_economic_renewal/007.php Coninunity
and Rural Development Institute, Cornell University.

[87] M.F. Neuts, Matrix-Geonietric Solutions in Stochastic Models, Johns Hopkins
University Press, Maryland, 1981.

[88] J.M. Nguyen, P. Six, T. Ch1 II1-- Il.-t, D. Antonioli, P. Lonthrail, P. Le Beux, An
objective method for bed capacity planning in a hospital department A comparison
with target ratio methods, Methods of information in medicine 46 (4) (2007)
:399-405.

[89] W.P. Pierskalla, Health care delivery, Presented at the National Science Foundation
Workshop on Engineering the Service Sector, Atlanta, 2001.









[90] W.P. Pierskalla, D. Brailer, Applications of operations research in health care
delivery, in Beyond the profit motive: public sector applications and methodology,
Handbooks in OR&MS, S. Pollock, A. Barnett, M. Rothkopf (eds), Vol 6,
North-Holland, New York, 1994.

[91] W.P. Pierskalla, D. Wilson, Review of operations research improvements in patient
care delivery systems, Working paper, University of Pennsylvania, Philadelphia,
1989.

[92] S. Rahman, D.K(. Smith, Use of location-allocation models in health service
development planning in developing nations, European Journal of Operational
Research 123 (2000) 437-452.

[93] J.F. Repede, J.J. Bernardo, Developing and validating a decision support system for
locating emergency medical vehicles in Louisville, K~entucky, European Journal of
Operational Research 75 (1994) 567-581.

[94] J.C. Ridge, S.K(. Jones, M.S. Nielsen, A.K(. Shahani, Capacity planning for intensive
care units, European Journal of Operational Research 105 (1998) 346-355.

[95] S.M. Ryan, Capacity expansion for random exponential demand growth with lead
times, Management Science 50 (2004) 740-748.

[96] F. Sainfort, Where is OR/ ilS in the present crises in health care delivery? Presented
at the Institute for Operations Research and the Management Sciences Annual
Meeting, Miami, 2001.

[97] D.P. Schneider, K(.E. K~ilpatrick, An Optimum Manpower Utilization Model for
Health Maintenance Organizations, Operations Research 23 (1975) 869-889.

[98] L. Shi, D.A. Singh, Essentials of the US Health Care System, Jones and Bartlett
Publishers, Maryland, 2005.

[99] R.A. Shumsky, Approximation and analysis of a queueing system with flexible and
specialized servers, OR Spectrum, Special Issue on Call Center Management 26 (3)
(2004) 307-330.

[100] D. Sinreich, Y. Marmor, Emergency department operations: the basis for developing
a simulation tool, IIE Transactions 37 (3) (2005) 233-245.

[101] V.L. Smith-Daniels, S.B. Schweikhart, D.E. Smith-Daniels, Capacity management
in health care services: review and future research directions, Decision Science 19
(1988) 899-918

[102] C. Smith, C. Cowan, A. So1.-, 1.1_ A. Catlin, Health Spending Growth Slows in 2004,
Health Affairs 25 (2006) 186-196.

[103] L. Snyder, Facility location under uncertainty: a review, IIE Transactions 38 (2006)
537-554.









[104] L. Snyder, 31. Daskin, Reliability Models for Facility Location: The Expected Failure
Cost Case, Transportation Science 39 (3) (2005) 400-416.

[105] D.A. Stanford, W.K(. Grassniann, The bilingual server system: a queueing model
featuring fully and partially qualified servers, INFOR 31 (4) (1993) 261-277.

[106] J. Thorson, Last time accessed: August 2006,
http://sepwww.stanford.edu/oldreports/sep2/01_b~t1

[107] United States Department of Labor Bureau of Labor Statistics, Last time accessed:
August 2006, http://www.bls. gov/oco/cg/cgs035 .htm#emply.

[108] 31. IUtley, S. Gallivan, K(. Davis, P. Daniel, P. Reeves, J. Worrall, Estimating bed
requirements for an intermediate care facility, European Journal of Operational
Research 150 (2003) 92-100.

[109] A.H. van Zon, G.J. K~oniner, Patient flows and optimal health-care resource
allocation at the macro-level: a dynamic linear progranining approach, Health Care
Alanagenient Science 2 (1999) 87-96.

[110] J.M.H. Vissers, Patient flow-based allocation of inpatient resources: A case study,
European Journal of Operational Research 105 (1998) 356-370.

[111] T.D. Vries, R.E. Beeknlan, Applying simple dynamic modelling for decision support
in planning regional health care, European Journal of Operational Research 105
(1998) 277-284.

[112] S. Yan, Approximating Reduced Costs under Degeneracy in a Network Flow
Problem with Side Constraints, Networks 27 (1996) 267-278.










BIOGRAPHICAL SKETCH

Chin-I Lin was born in Taipei, Taiwan. She received her B.S. and 1\.S in civil

engineering from the National Central University in Taiwan in 1994 and 1996, respectively.

From 1997 to 2002, she worked for Cl....! I Airlines, and her 1!! li r~ tasks included demand

forecast, market analysis, route analysis, and fleet planning. She pursued her master

and doctoral degrees in the Department of Industrial and Systems Engineering at

the University of Florida since 2002. Chin-I's main research interest is Operations

Research, and topics of special interest are health care management and airline flight/crew

scheduling. Thus far, her research has focused on capacity management in health care

delivery.





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Iwouldliketothankallpeoplewhohavehelpedandinspiredmeduringmydoctoralstudy.Iwanttoexpressmysinceregratitudetomydissertationadvisor,Dr.ElifAkcal,forherguidance,insightandsupportduringthisresearchandstudy.Iamalsogratefultomycommitteemembers,Dr.FaridAitSahlia,Dr.P.OscarBoykinandDr.SiriphongLawphongpanich,fortheirconstructivesuggestionsandcomments.Iwishtoextendmywarmestthankstomymentor,Dr.ShangyaoYan,forleadingmetotheeldofoperationsresearch,hisfriendshipandnumerousfruitfuldiscussions.Mydeepestgratitudegoestotomyparents,Zhe-XiongandFang-Xue,andmyhusband,Chung-Jui.Withouttheirunderstandingandencouragement,itwouldhavebeenimpossibleformetocompletemydegree. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 2AGGREGATEHOSPITALBEDCAPACITYPLANNING ............ 13 2.1Introduction ................................... 13 2.2LiteratureReview ................................ 15 2.3ProblemFormulation .............................. 17 2.3.1RestrictedBedCapacityPlanningProblem .............. 19 2.3.2RestrictedBedCapacityPlanningProblemwithShuttering ..... 22 2.4IllustrationoftheModel ............................ 25 2.4.1ARepresentativeDecision-MakingScenario .............. 25 2.4.2Experiment1{AnApplicationoftheModel ............. 26 2.4.3Experiment2{AssessingtheImpactofProblemParameters .... 29 2.5Extensions .................................... 33 2.6ConcludingRemarksandFutureResearchDirections ............ 34 3HEALTHCARETEAMCAPACITYPLANNING ................ 37 3.1Introduction ................................... 37 3.2LiteratureReview ................................ 42 3.3ProblemFormulation .............................. 44 3.4QueueingAnalysis ............................... 47 3.4.1PreemptiveCase:EmergencyMedicineServices ........... 48 3.4.2Non-PreemptiveCase:OutpatientClinicServices .......... 51 3.5ComputationalStudy .............................. 56 3.5.1ComputationalPerformanceoftheDBAMethod ........... 57 3.5.2ComputationalPerformanceoftheHCTSCPModel ......... 62 3.6ConcludingRemarksandFutureResearchDirections ............ 66 4HOSPITALBEDALLOCATIONPROBLEM ................... 69 4.1Introduction ................................... 69 4.2LiteratureReview ................................ 70 4.3ProblemFormulation .............................. 73 4.4SolutionAlgorithms .............................. 75 5

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............................ 75 4.4.2GreedyRandomizedAdaptiveSearchProcedure ........... 80 4.4.3HybridizationofGA&GRASP .................... 81 4.5ComputationalStudy .............................. 83 4.5.1SummaryofResultsObtainedbyGA ................. 85 4.5.2SummaryofResultsObtainedbyGRASP .............. 88 4.5.3SummaryofResultsObtainedbyHA ................. 88 4.6ConcludingRemarksandFutureResearchDirections ............ 92 5EMERGENCYROOMSERVICESFACILITYLOCATIONANDCAPACITYPLANNING ...................................... 94 5.1Introduction ................................... 94 5.2LiteratureReview ................................ 95 5.3ProblemFormulation .............................. 97 5.4SolutionApproach ............................... 100 5.4.1LagrangianRelaxationApproach ................... 100 5.4.2LowerBound .............................. 101 5.4.3UpperBound .............................. 103 5.4.4LagrangianMultipliers ......................... 106 5.5ComputationalStudy .............................. 107 5.5.1ExperimentalDesign .......................... 107 5.5.2ExperimentalResults .......................... 108 5.6ConcludingRemarksandFutureResearchDirections ............ 111 6CONCLUSIONSANDFUTURERESEARCHDIRECTIONS .......... 114 REFERENCES ....................................... 117 BIOGRAPHICALSKETCH ................................ 125 6

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Table page 2-1Parametersettingsforthebasescenario,S1 .................... 27 2-2Scenariodescriptionsforexperiment1 ....................... 27 2-3SummarystatisticsfortheRBCPwSproblem'ssolutiontime(inCPUseconds)asafunctionofinitialeectivecapacity ...................... 31 2-4SummarystatisticsfortheRBCPwSproblem'ssolutionasafunctionofcapacitylevelsandthelengthoftheplanninghorizon .................... 32 3-1Thepossibletransitionsenterstate(n1;n2;m)fortheEDsetting ........ 49 3-2ComputationalrequirementoftheDBA,AGE,andGEmethods ......... 58 3-3Relativeandpercentageerror:preemptivecase .................. 60 3-4Relativeandpercentageerror:non-preemptivecase ................ 61 3-5Teamcongurations ................................. 63 3-6Parametersettings .................................. 63 3-7Impactofthefractionofclass1patientsonthesimilarityindex ......... 65 3-8Impactofunitpatientdelaycostonthesimilarityindex ............. 65 3-9Impactofmaximumallowableaveragetimeinsystemonthesimilarityindex .. 65 4-1Near-optimalsolutionsobtainedbyusingCPLEX ................. 86 4-2SolutionsobtainedbyGA .............................. 87 4-3SolutionsobtainedbyGRASP ............................ 89 4-4SolutionsobtainedbyHA .............................. 91 5-1Experimentalfactorsettings ............................. 107 5-2Experiment1:eectsofmaximumnumberoffacilitiesopened .......... 109 5-3Experiment2:eectsofcapacitysetting ...................... 110 5-4Experiment3:eectsofdiversionprobability ................... 110 5-5Experiment4:eectsoftimevalue ......................... 111 5-6Performanceoftheheuristic ............................. 112 7

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Figure page 2-1NetworkowrepresentationforRBCPwithc0=300,B=25,n=1,andT=4 ... 21 2-2NetworkowrepresentationforRBCPwSwithc0=275,B=25,n=1,andT=4 24 2-3Patientarrivalrateforexperiment1 ........................ 28 2-4Optimalcapacityplansforexperiment1 ...................... 28 2-5Numberofnodesinthenetworkasafunctionofinitialeectivebedcapacity .. 30 3-1AnillustrationofthenetworkrepresentationforHCTSCP ............ 47 3-2Two-dimensionalCTMCapproximation ...................... 52 4-1Pseudo-codeofthegeneticalgorithm ........................ 76 4-2Pseudo-codeofthepopulationgeneratingprocedure ................ 77 4-3Pseudo-codeofoccupancy-drivenallocation .................... 77 4-4Pseudo-codeofrandom-rectiedprocedure ..................... 78 4-5Pseudo-codeofcrossoverprocedure ......................... 79 4-6Exampleofcrossover ................................. 79 4-7Pseudo-codeofthemutationprocedure ....................... 80 4-8Pseudo-codeofGRASP ............................... 80 4-9Pseudo-codeofgreedyrandomizedconstructionprocedure ............ 82 4-10Pseudo-codeoflocalsearchprocedure ........................ 83 4-11Pseudo-codeofHA .................................. 83 4-12Pseudo-codeofelitesetgenerationprocedure ................... 84 4-13Theupdatingfunctionsof 90 5-1Pseudo-codeoftheLagrangianrelaxation ..................... 101 5-2Pseudo-codeofthefeasiblesolutiongeneration ................... 104 5-3Pseudo-codeofcoveringalldemandnodes ..................... 105 5-4Pseudo-codeofclosingfacilities ........................... 105 5-5ConvergencespeedofthemodiedLR ....................... 109 8

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Healthcarecapacityplanningistheartandscienceofpredictingthequantityofresourcesrequiredtodeliverhealthcareserviceatspeciedlevelsofcostandquality.Becauseofvariabilityinthearrivalofpatientsandinthedeliveryofhealthcareservices,successfullymeetingthedemandforhealthcareservicesisadauntingtaskthatrequiresanunderstandingoftheinherenttrade-obetweenitscostandqualityofservice. Inourwork,wemodelthegeneralhealthcaresystemsasqueueingstationsandincorporatequeueingtheoryintoanoptimizationframework.Thequeueingmodelingapproachcapturesthestochasticnatureofarrivalsandservicetimesthatistypicalinhealthcaresystems.Theoptimizationframeworkdeterminestheminimumcostcapacityrequiredtoachieveatargetlevelofcustomerservice.Theinclusionsofqueueingequationsanddiscretecapacityoptionsresultthecapacityplanningmodelsinnon-linearintegerprogrammingformulations. Wedevelopeectivesolutionalgorithmstoobtainhighqualitysolutionsparticularlyforrealistic-sizedproblems.Fortheanalysisofunderlyingqueuingsystems,weeitheruseavailableresultsfromtheliteratureordevelopapproximations.Forthesolutionofoptimizationmodels,weemploynetworkoptimization,meta-heuristic,andLagrangianrelaxationapproachestodevelopeectivesolutionalgorithms.Wepresentresultsfromextensivecomputationalexperimentstodemonstratethecomputationaleciencyandeectivenessoftheproposedsolutionapproaches. 9

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Capacityplanningdecisionsareimportanttoanyindustry,especiallytohealthcareindustrybecausenotonlyitrelatestothemanagementofhighlyspecializedandcostlyresources(i.e.,nurses,doctors,andadvancedmedicalequipment),butalsoitmakesadierencebetweenlifeanddeathincriticalconditions.Healthcarecapacityplanninginvolvespredictingthequantityandparticularattributesofresourcesrequiredtodeliverhealthcareserviceatspeciedlevelsofcostandquality. Accordingtotheresourceavailabilities,capacityplanningcanbeclassiedintothreelevelsincludingstrategic,tacticalandoperationallevels.Inthestrategiclevel,decisionmakersfocusonthelongtermcapacitydecisionssuchaslocationsofmedicalfacilitiesorsizesofmedicalfacilitiesandworkforce.Thetacticallevelcapacityplanconcernspolicieswhichcouldimprovetheserviceperformance,ontheperspectiveofeitherhealthcareprovidersorreceivers,byreallocating,expandingordownsizingthecurrentresources.Incontrast,thecapacityplanontheoperationallevelconcentratesonhowtomeetdemandbyusingtheexistingresourcesthroughappropriatemethodssuchasschedulingoroverrun. Inourstudy,wefocusonlong-termlevelhealthcarecapacityplanningdecisions,wheretheknowledgeoftheperformanceofthesystematsteadystateissucient.Inaddition,wemodelthehealthcareservicefacilitiesasqueueingsystemstotakethestochasticpatientarrivalsandlengthofstayintoaccountandmeasurethesystemperformance.Thepurposeofourstudyistoincorporatequeueingmodelsintooptimizationframeworktodeterminetheoptimalcapacitylevelthatminimizescostwhilemaintainingadesiredlevelofperformanceonpatients'servicequalityandnancialand/oroperationalrestrictions. Inourwork,westudyfourdierent,practicallyrelevant,long-termcapacityplanningproblems.Chapter 2 introducestheaggregatehospitalbedcapacityplanning(AHBCP) 10

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16 17 ]toestimatethepatients'expectedwaitingtime,wedeterminetheoptimalbedcapacityplanoveraniteplanninghorizonutilizinganetworkowapproach. Chapter 3 presentsthehealthcareteamcapacityplanning(HCTCP)problem,whichndsapplicationsinhospitalemergencyroomandoutpatientclinicalsettings.TheunderlyingqueueingsystemismorecomplexthantheoneusedforAHBCP.Inparticular,weconsiderasystemwithtwoclassesofpatientswithdierentprioritiesandtwotypesofhealthcareteamswithpatientclass-dependentservicerates.Moreover,thereisasymmetricsubstitutabilitybetweentheteams,i.e.,whileoneteamcanprovideservicetobothclassesofpatient,whereastheotherteamcanserveonlyoneclassofpatient.Forthisqueueingsystem,wedevelopanapproximationapproachtocomputetheaveragetimethatapatientspendsinthesystemforeachpatientclass.Then,weintegratetheresultsfromapproximationmethodintoanoptimizationmodeltomakelong-termhealthcareteamcapacitydecisions. Chapter 4 statesthehospitalbedallocation(HBA)problem,whichisanextensiontheAHBCPproblem.Aftertheaggregatebedcapacityisspecied,thenextstepinvolvedisconcernedwiththeallocationofaggregatebedcapacityamongdierentmedicalcareunits(MCUs).WemodeleachMCUinahospitalasaM=M=c=cqueueingsystemtoestimatetheprobabilityofrejectionwhentherearecbedsintheunit.WedevelopanoptimizationmodeltoallocatetheaggregatebedcapacityacrossdierentMCUs. Chapter 5 detailstheemergencyroomservicesfacilitylocationandcapacityplanning(ERSFLCP)problem,inwhicheachfacilityismodeledasaM=M=c=cqueueingsystem.Weconstructafacilitylocationmodel,whichsimultaneouslydeterminesthenumberoffacilitiesopenedandtheirrespectivelocationsaswellasthecapacitylevelsofthefacilitiessothattheprobabilitythatallserversinafacilityarebusydoesnotexceedapre-determinedlevel. 11

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6 providesasummaryofthefourproblemswehaveinvestigatedandsuggestsfutureresearchdirections. 12

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Whilecapacityplanninghaschallengedhealthcaredecisionmakersandresearchersfordecades[ 90 91 101 ],thereisarenewedsenseofurgencytoaddressthisproblem.Inadditiontotheperennialstrugglebetweenthecontinuallyincreasingcostsofhighlyspecializedandscarceinputs(i.e.,skilledandexiblesta,advancedclinicalandmedicaltechnologyandequipment,physicalspaceandsupplies)anddeclininggovernmentandprivatereimbursements[ 89 96 ],thedemandforinpatientcarehasbeengrowingsubstantially.AccordingtotheAmericanHospitalAssociation(AHA),whileaveragelengthofstay(ALOS)remainedunchangedat5.7days,allcommunityhospitalvolumestatisticsincreasedfrom2002to2003:inpatientadmissionsby0.9%to34.8million, 13

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7 ].However,thenumberofhospitalsofalltypesdecreasedby30to5,764,therewere32fewercommunityhospitals,and8,000fewercommunityhospitalbedsin2003[ 7 ]. Inthispaper,wefocusonaggregatehospitalbedcapacityplanningdecisions.Wedevelopamodeltosimultaneouslydeterminethetimingandmagnitudeofchangesinbedcapacitythatminimizescapacitycost(includingthecostofchangingcapacityaswellasthecostofoperatingcapacity)whilemaintainingadesiredleveloffacilityperformance(e.g.,limitingapatient'sexpecteddelaybeforebeingadmittedtoabedandkeepingexpenseswithinbudget)overaniteplanninghorizon.Wedividetheplanninghorizonintodiscretetimeperiodsofequallength,andassumethatthesystemachievessteadystateineachoftheseintervals.Thisallowsustousequeuingmethodologytoanalyzesystemperformance,butthistypicallyleadstononlinearequationsinourformulation.Ashospitalbedcapacitymustbeintegervalued,ourplanningmodelisalarge-scalenonlinearintegeroptimizationmodelthatminimizestotalcostwhileachievingatargetedlevelofsystemperformance.Weshowthatsomepracticalconsiderationsleadtosimplicationsinthemodel,whichleadstoanetworkowformulationfortheproblemthatcanbesolvedinpolynomialtime. Avarietyofproblemsthatariseinthecontextoftransportation,nance,manufacturing,andservicesystemscanbemodeledasnetworkowmodels[ 2 ].Anetworkisacollectionof(capacitatedoruncapacitated)nodesand(directed/undirectedandcapacitated/uncapacitated)arcs,wherethearcslinkonenodetoanotherandcarryowfromonenodetoanother.Well-knownnetworkowmodelsaretheshortestpath,maximumow,andminimumtotalcostowformulations,forwhichecientsolutionalgorithmsexist[ 2 ].Inourwork,weshowthatthecapacityplanningmodelweconsidercanbetransformedintoashortest 14

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Theremainderofthispaperisorganizedasfollows.Section 2.2 providesabriefoverviewofthehistoryandcurrentresearchinhospitalbedplanning.InSection 2.3 ,wedescribethesystemandgivethreemodelsforplanninghospitalbedcapacity.InSection 2.4 ,usingdatafromamedium-sizedmedicalcenter,weprovideacomputationalstudytoillustratehowthemodelformulationscanbeusedandhowchangesinproblemparameterscanaectourabilitytoobtainanoptimalsolution.Section 2.5 ,oersseveralpracticalextensionsofourmodel.Last,wegiveconcludingremarksanddiscussfutureresearchdirectionsinSection 2.6 10 15 29 45 46 50 ].Lessthanadecadelater,inpartduetorenewedgrowthindemandforinpatientservices[ 7 31 ],mosthospitalsarecurrentlyfacingconsiderablespaceandresourcerestrictionsforcingthemtocontemplateexpensiverenovationsand/ornewconstructionprojectstoincreasebedcapacity[ 11 31 57 ].However,whetherhospitals,infact,needtheadditionalcapacityappearstobeunresolved[ 10 45 ].Ononehand,increasedinpatientadmissionscoupledwithfewerhospitalsandfewerhospitalbedswouldsupporttheargumentinfavorofcapacityincreases[ 7 31 ].Conversely,levelordecreasingaveragelengthofstayandacorrespondingdecreaseintheaverageinpatientoccupancyratemayimplythatexistingcapacityissucient[ 10 ].Regardless,determiningtheoptimalnumberandorganizationofhospitalbedscontinuestobeachallenge. Theabilitytoanticipatebeddemandandmatchitwiththeappropriatebedsupplyiscriticaltoeectivebedplanning.Healthcaredecisionmakersknowthatbothwillbeinuencedbyanumberoffactors.Factorsinternaltothedecisionmakersinclude 15

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45 46 ].Externally,factorsfacingdecision-makersincludeatypicalchangesincommunityhealth(e.g.,severeustrains),annualholidays(e.g.,Thanksgiving),andtheavailability,size,andcompositionofappropriatemedicalpersonnel.Historically,startingwiththeHill{BurtonActof1946,bedcapacityplanninghastendedtobebasedontargetoccupancylevels(TOLs)thatareassumedtoreectcapacitylevelsthatachieveanappropriatebalanceofcost,patientdelays,andresourceutilization.TOLsarederivedusinganalyticmodelsoftypicalhospitalsindierentcategoriesandarebasedonacceptablepatientdelaysfordierentservices.However,GreenandNguyen[ 46 ]usequeuingmodelstoinvestigatetherelationshipbetweenoccupancylevelsanddelay,andconcludedthatusingTOLsastheprimarydeterminantofbedcapacityisinadequateandmayleadtoexcessivedelaysforbeds.Inparticular,aTOLdoesnotnecessarilycorrespondtoadesiredservicelevel,andthereisaneedtoquantifythedesiredservicelevelandmeasureitscostimplicationsaccurately. Ryan[ 95 ]providesacapacityexpansionmodelwithexponentialdemandandcontinuoustimeintervalsandcontinuousfacilitysizes.Inthecontextofhealthcareplanning,however,itismorerealistictomodelcapacityexpansionastheproductoflimited,discretechoicesasroutineplanningsessions(e.g.,bimonthlyorquarterly)wherecapacityincreasesordecreasesoccurinsomexedbedamountsuchasa20-bedunit.BretthauerandC^ote[ 26 ]modelageneralhealthcaredeliverysystemasanetworkofqueuingstationsandincorporatethequeuingnetworkintoanoptimizationframeworktodeterminetheoptimalcapacitylevelssubjecttoaspeciedlevelofsystemperformance 16

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7 ],theactualtforagivenfacilitywillnotbeknownuntilthedemandpresentsitself.Therefore,forthepurposesofcapacityplanning,tcanbeforecastedbyaseasonallyadjustedtrendline,forexample[ 32 ].Lettdenotethemaximumallowableexpecteddelayforapatientbeforethepatientisadmittedtoabedinperiodt.Wenotethatthenumberofbedsinthesysteminagivenperiodcanbelimitedduetootherresourcelimitationsincludingasthephysicalsizeofthefacilityand/ortheamountandtypeofpersonnelavailable.Letc0betheinitialbedcapacityinthehospital.Last,thereisabudgetlimitontheamountofmonetaryresourcesthatcanbeallocatedtopurchasingadditionalbedcapacitydenotedbyt. Wehavethreetypesofdecisionvariables.Letxtbenumberofbedsinperiodt.Letx+tbetheamountofincreaseinbedcapacityatthebeginningofperiodt,andxtthe 17

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minTXt=1f(xt;t;t)+TXt=1g(xt1;xt)+TXt=1h(xt) (2{1)subjecttow(xt;t;t)t8t Theobjectivefunction( 2{1 )minimizesthetotalcostofpatientwaiting,changingthebedcapacity,andoperatingtheexistingbedcapacity.Constraint( 2{2 )imposesamaximumallowablelimitontheexpectedpatientwaiting.Forexample,inordertoquantifytheexpecteddelayforapatienttobeadmittedtoabed,weassumethatthehospitalcanberepresentedasaGI=G=squeueingsystemandusetheexpectedwaitingtimeapproximationprovidedbyBitranandTirupati[ 16 17 ]tocalculateapatient'sexpectedwaitforahospitalbed.Constraint( 2{3 )setstheinitialbedcapacitywhileconstraint( 2{4 )isaowbalanceequationstatingthatthenumberofbedsavailableinaperiodisequaltothenumberofbedsavailableinthepreviousperiodplustheincreaseinbedcapacityminusthedecreaseinbedcapacity.Constraint( 2{5 )isthebudgetconstraint 18

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2{6 )ensuresthatthenumberofbedsavailableandchangesinbedcapacityareintegervalued. 7 ].Inpractice,bedcapacityisincreasedordecreasedinbatches,andistypicallychangedinintegermultiplesofabasevalue,say,inmultiplesof10or25correspondingtothesizeofaunit.Asaresult,thereareonlyalimitednumberofchoicesforchangingcapacityineachperiod.Therefore,constraintsthatcapturethechangeincapacitycanbereplacedbyasetofdiscretealternativeconstraints,requiringthatonlyonealternativeischoseninthesolutionforeachperiod.Then,theoriginalnon-linearintegerprogrammingproblembecomesanonlinearbinary(i.e.,zeroone)integerprogrammingproblem,whichwerefertoastherestrictedbedcapacityplanning(RBCP)problem. IntheRBCPproblem,wearegivenabasevalueofBinmultiplesofwhichthebedcapacitycanbeincreasedordecreasedandweletnbethenumberofpossibledistinctlevelsofcapacityincreaseordecrease.Thatis,givenbedcapacitycinperiodt,thebedcapacityinperiodt+1canbeoneof(cnB)+,(c(n1)B)+,...,(cB)+,c,c+B,...,c+(n1)B,c+nB,where(x)+=maxf0;xg.Weassumethatallacquirednewadditionalcapacityisavailableandbecomeseectivecapacityinthesameperiod.Letz+it=1iftheavailablebedcapacityisincreasedbyiBatthebeginningofperiodtfori=1,2,...,n;and0otherwise.Similarly,letzit=1ifthebedcapacityisdecreasedbyiBatthebeginningofperiodtfori=1,2,...,n;and0otherwise.WecannowformulatetheRBCPproblemasanonlinearzerooneintegerprogrammingproblemasfollows: minTXt=1f(xt;t;t)+TXt=1g(xt1;xt)+TXt=1h(xt) (2{7)subjectto 19

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AsintheAHBCPproblem,objectivefunction( 2{7 )minimizesthetotalcostofpatientdelay,changingthebedcapacity,andoperatingtheexistingbedcapacity,constraint( 2{8 )imposesamaximumallowablelimitontheexpectedpatientdelay,constraint( 2{9 )setstheinitialbedcapacity,andconstraint( 2{10 )isaowbalanceequation.Constraint( 2{11 )ensuresthatonlyonechoiceforchangingthecapacityisallowedineachperiod.Constraint( 2{12 )imposesthebudgetconstraintontheamountofmoneyallocatedtochangingbedcapacity.Constraints( 2{13 )and( 2{14 )ensurethenonnegativityofthebedcapacitylevelandcapacitylevelselectiondecisionvariables,respectively. AnattractivefeatureoftheRBCPproblemisthatanetworkrepresentationcanbedeveloped.ConsideraT-partitegraphwithTlayerseachrepresentingatimeperiodt=1,2,...,Tintheplanninghorizon.Let(t;c)denotethesystemwhentherearecbedsinperiodt.LetC(c)bethesetofreachablecapacitylevelsinthenextperiodifthecapacityinthecurrentperiodisc,andwehaveC(c)=f(cnB)+;(c(n1)B)+;:::;(cB)+;c;c+B;:::;c+(n1)B;c+nBg.LetStbethesetofallcapacitylevelsreachableinperiodtfromallcapacitylevelsinperiodt1.Letdsbeasupercialsourcenodeconnectedonlytonode(0;x0)withzeroarclength.Node(0;x0)representsthebeginningstatewheretherearex0bedsinthehospitalattimet=0.Let(0;x0)beconnectedtoallnodes(1,x')forx'2C(x0).Ifw(x';1;1)1(i.e.,thepatientwaitingtimeconstraintisnotviolated)andg(x0;x')1(i.e.,thebudgetconstraintisnotviolated),thenthe 20

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Figure 2-1 providesanexampleofthenetworkrepresentationfortheRBCPproblemwherec0=300,B=25,n=1,andT=4.Inthisgure,apathfromthesupercialsourcenodetothesupercialsinknoderepresentsaplanforthebedcapacityovertheplanninghorizon.TheshortestpathwithoutcontaininganyarcwithcostMyieldsthecapacityplanwithtotalminimumcostthatobeysthepatientwaitingtimeandbudgetconstraintsovertheplanninghorizon.Ifnosuchpathcanbefound(i.e.,theshortestpathcontainsatleastonearcwithcostM),thentheproblemisinfeasibleandnocapacityplanthatobeysthewaitingtimeandbudgetconstraintsovertheplanninghorizoncanbefound. Figure2-1. NetworkowrepresentationforRBCPwithc0=300,B=25,n=1,andT=4 21

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2 ]. AswiththeRBCPproblem,anetworkrepresentationcanbedevelopedfortheRBCPwSproblem.ConsideraT-partitegraphwithTlayerseachrepresentingatime 22

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Figure 2-2 providesanexampleofthenetworkrepresentationfortheRBCPwSproblemwherec0=275,c0=300,B=25,n=1,andT=4.Foreaseofexposition,thethinarcsrepresentopening,maintaining,orshutteringofexistingcapacity,whereasthethickarcsrepresenttheacquisitionofnewcapacity.Inthisnetwork,apathfromthesupercialsourcenodetothesupercialsinknoderepresentsaplanforthebedcapacitythroughout 23

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Figure2-2. NetworkowrepresentationforRBCPwSwithc0=275,B=25,n=1,andT=4 RecallingtheRBCPproblem,wehavespeciedthenumberofarcsandnodesinthenetworktodeterminethetimetoobtaintheoptimalsolution.However,fortheRBCPwSproblem,sinceexistingcapacitycanbeincreasedfurtherthroughcapitalacquisition,theanalysisbecomesslightlymorecomplicatedanddependentontheinitialstate(i.e.,theamountofeectiveandexistingbedcapacity).Ifnoadditionalcapacity 24

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2 ]. Thisfacilitywouldliketodetermineanoptimalbedcapacityplanforthenexteightquarters,correspondingtoitsoperational,budgetary,andstrategicplanningperiods.Becausecapacityplanningmayinvolveasubstantialcapitalcommitment,itisimperativethatanycapacityexpansionplanbecarefullydevelopedandjustiedbaseduponthe 25

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2.4.1 ,andwerefertoitasS1.Table 2-1 liststherelevantparametersettingsforS1,andotherexperimentalscenariosrelativetothisscenarioaregiveninTable 2-2 Attheoutset,weprovideanestimatedrangeofdemandforthefacilityovertheplanninghorizon.Normally,asingleseasonallyadjustedtrendlinewouldbecomputedtoforecastthepatientarrivalratebasedonhistoricdemanddata.Instead,toillustratetheextentofvariationindemand,Figure 2-3 displaysasetofsimulatedpatientarrivalratesovertheplanninghorizonbaseduponthescenariosgiveninTable 2-2 .Wenotethatsomeoftheparameterchangesdirectlyimpactthepatientarrivalrate,anddierentpatientarrivalratesaregenerated.InS1,S3,S4,S5,andS6,thechangedparametersdonotimpactthearrivalratefunction,sothesescenarioshaveidenticalarrivalrates.(Note 26

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Parametersettingsforthebasescenario,S1 ParameterValue LengthoftheplanninghorizonT=8quarters,t=1,2,...,8 Forecasteddemandpertimeperiodt Initialexistingbedcapacityc0=350 Initialeectivebedcapacityc=350 Numberoflevelsofcapacityincreaseordecreasen=2 IncrementalamountofcapacitychangeB=10 Costtooperateaneectivebed$2,000/bed Costtoshutteraneectivebed$2,500/bed Costtoreactivateashutteredbed$2,500/bed Costtoacquireanewbed$200,000/bed (i.e.,expandcapacitythroughcapitalinvestment) Coecientofvariationforarrivalscat=0:5 Coecientofvariationforservicecst=0:5 Maximumexpecteddelayperpatientat=1hour Costofwaiting$300/hour Serviceratet=15:8patientsperbed Table2-2. Scenariodescriptionsforexperiment1 ScenarioDescriptionParameterchange S0Leveldemandb=0S1BasescenarioS2Increasedrateofdemandb=256S3Higherdemandvariabilitycat=2.0S4Higherservicevariabilitycst=2.0S5Highercostofwaitingperpatient$1,200/hourS6Smallermaximumexpecteddelayperpatientt=0.25ofanhour thatwithS3,highervariabilityinthearrivalrateimpactstheperformanceconstraintforaveragewaitingtime,notthearrivalratefunction.)ForS0andS2,thepatientarrivalratefunctionhasnotrendandahighertrendcomparedtoS1,respectively.Hence,arrivalratesgeneratedforthesescenariosaresignicantlydierentfromeachotherandS1. WehaveimplementedournetworkowapproachusingtheC++programminglanguageandsolvedforthescenariosusingapersonalcomputerwith3.0GHzPentiumIV 27

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Patientarrivalrateforexperiment1 processorand512MBRAMmemory.WeobtainedtheoptimalsolutionforeachscenarioandtheresultsaredepictedinFigure 2-4 ,whereeachlinerepresentstheoptimalcapacityplanthatcorrespondstooneofthesevenscenarios. Figure2-4. Optimalcapacityplansforexperiment1 28

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2-4 ,wehavethefollowingobservations.ForS1,werstobserveageneralreductioninthebedcapacity,thenagradualincreaseneartheendoftheplanninghorizon.Theinitialbedcapacityseemstobehigherthanneeded,andasaresult,thebedcapacityisreducedtoreducetotalcostsovertheplanninghorizonwhilemaintainingtheaveragewaitingtimeconstraint.Ofcourse,whenthedemandincreasesduetotheunderlyingtrend,thebedcapacityisincreased.WhendemandislevelasinS0,alowerenvelopeisformedrelativetothebasecase(i.e.,thebedcapacityforS0islessthanorequaltothebasecase).Similarly,withanincreasedrateofdemandasinS2,anupperenvelopeisformedrelativetothebasecase.WithincreasedvariationasinS3andS4,theoptimalcapacityplansaresimilartoS1'scapacityplanbuttendtorequirehighercapacitywhenthearrivalrateisincreasing.Whenthearrivalrateincreasesinperiods6,7,and8,becausethehigherarrivalvariabilityandhigherservicevariabilityaecttheaveragewaitingtimeconstraint,morecapacityisrequiredtokeepfromviolatingthisperformanceconstraint.Likewise,withahighercostofwaitingperpatientasinS5oratighteraveragewaitingtimeperformanceconstraintasinS6,theoptimalcapacityplanstendtorequirecapacityslightlyhigherthanthebasecase.Notsurprisingly,thenetresultofthisexperimentindicatesthatoptimalbedplansaredrivensubstantiallybychangesindemand.Whilehealthcaredecisionmakersmaynotbeabletoaectoveralldemandfortheirservices,iftheycanreducevariabilityinarrivals[ 78 ]orarewillingtotoleratealessstringentperformanceconstraint,lesscapacitywillberequired. 29

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Intherstpartofthisexperiment,wexthenumberoflevelstovarybedcapacityandthedurationoftheplanninghorizoninadditiontosomeotherproblemparametersconstantandexaminetheimpactofdierentratiosofexistingtoeectivebedcapacity.Usingtheassumptionsforthebasecasescenario,S1,fromthepreviousexperiment,weconsidertendierentlevelsoftheeectivebedcapacityintheinterval[260,350].Wegenerated30randomtestinstancesforeachoftheselevelsandthesummaryresultsareprovidedinFigure 2-5 andTable 2-3 Figure2-5. Numberofnodesinthenetworkasafunctionofinitialeectivebedcapacity InFigure 2-5 wedepictthenumberofnodesinthenetwork,andinTable 2-3 wereportthetimetobuildthenetworkandtimetoobtainthesolutionforeachleveloftheinitialeectivebedcapacity.Thenumberofnodesincreasesastheratioofeectivebedcapacitytoexistingbedcapacityapproachesone,andthisbehaviorisclearlydepictedinFigure 2-5 .However,inTable 2-3 ,weseethatanincreaseinthesizeofthenetwork 30

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Table2-3. SummarystatisticsfortheRBCPwSproblem'ssolutiontime(inCPUseconds)asafunctionofinitialeectivecapacity ofeectiveBuildthenetowrkObtainthesolution(inCPUseconds) bedcapacityMin.Avg.Max.Min.Avg.Max.Min.Avg.Max. 2600.00.00.20.00.00.00.00.00.22700.00.00.10.00.00.00.00.00.12800.00.00.10.00.00.00.00.00.12900.00.00.10.00.00.00.00.00.13000.00.00.10.00.00.00.00.00.13100.00.00.10.00.00.00.00.00.13200.00.00.10.00.00.00.00.00.13300.00.10.10.00.00.00.00.10.13400.10.10.10.00.00.00.10.10.13500.10.10.10.00.00.00.10.10.1 2-4 FromTable 2-4 ,asthenumberoflevelsofcapacitychangeandthelengthoftheplanninghorizonincreases,thenumberofnodesinthenetworkincreases.Theincreaseinthenumberofnodesimpactsthetotaltimerequiredtoobtaintheoptimalsolution.However,acloserexaminationoftheresultsrevealstheincreaseinthenumberofnodesinthenetworkhasadirectimpactonthetimerequiredtobuildthenetwork,andhasalmostnoimpactonthetimetoobtainthesolution.Onlyinthesettingwiththelargesttestinstances(i.e.,n=5andT=16)doweobserveanincreaseinthetimetoobtainthe 31

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SummarystatisticsfortheRBCPwSproblem'ssolutionasafunctionofcapacitylevelsandthelengthoftheplanninghorizon NodesinthenetworkArcsinthenetworkBuildthenetworkObtainthesolutionAvg.(Min.,Max.)Avg.(Min.,Max.)Avg.(Min.,Max.)Avg.(Min.,Max.)Avg.(Min.,Max.) (2,8)323.7(194,490)1,213.9(737,1,865)0.04(0.02,0.14)0(0,0)0.04(0.02,0.14)(3,8)707.1(451,982)3,566.3(2,226,5,061)0.13(0.06,0.2)0(0,0.02)0.13(0.06,0.2)(4,8)1,257.5(858,1,642)7,989.3(5,273,10,665)0.31(0.17,0.44)0(0,0.02)0.31(0.17,0.44)(5,8)1,980(1,455,2,470)15,199(10,834,19,349)0.64(0.41,0.89)0(0,0.02)0.64(0.41,0.89)(2,12)997.7(662,1,338)4,067.3(2,661,5,521)0.19(0.09,0.34)0(0,0.02)0.19(0.09,0.34)(3,12)2,296.2(1,697,2,858)12,850.4(9,328,16,195)0.63(0.38,0.86)0(0,0.02)0.63(0.39,0.86)(4,12)4,165.3(3,278,4,950)29,679(22,957,35,669)1.72(1.16,2.25)0(0,0.02)1.72(1.16,2.25)(5,12)6,607.5(5,469,7,614)57,209.8(46,688,66,583)4.17(3.02,5.31)0(0,0.02)4.17(3.02,5.31)(2,16)2,468.2(1,574,3,282)10,586.7(6,637,14,225)0.62(0.31,0.99)0(0,0.02)0.62(0.31,0.99)(3,16)5,659.9(4,110,6,954)33,550.5(23,969,41,609)2.25(1.36,3.36)0(0,0.02)2.25(1.36,3.36)(4,16)10,211.9(8,022,11,986)77,310.6(59,925,91,473)6.83(4.83,8.84)0(0,0.02)6.83(4.83,8.84)(5,16)16,128(13,278,18,378)148,628.7(120,987,170,537)57.57(13.13,138.22)0.49(0,1.25)58.06(13.13,139.28)

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Inourmodel,wetreattheperformanceconstraintsasahardconstraint.Thatis,ifaparticularcapacitylevelviolatestheperformanceconstraint,thenasolutionwiththatparticularcapacitylevelisnotfeasible,andisdroppedfromfurtherconsideration.However,theperformanceconstraintcanbemodeledasasoftconstraintwherewecandeliberatelyallowtheviolationoftheperformanceconstraintwhileincurringapenaltycosttobeaddedtotheobjectivefunction.Wecanjustifythisconstraintbynotingthatlagstypicallyexistbetweencapacitylevelssotheremightbeperiodsoftimewherethefacilityisoperatingaboveitstypicalutilizationandthecapacityexpansioncannotoccurquicklyenoughtoallowtheorganizationtoreacttothechangeindemand.Toillustratehowourmodelcanbereformulatedwiththesoftconstraint,letvtbetheamountoftheviolation,stbetheamountofslackintheperformanceconstraint,and(vt)bethepenaltycostincurredforviolatingtheperformanceconstraintinperiodt.Then,consideringtheRBCPproblem,objectivefunction( 2{7 )wouldbereplacedwith:minTXt=1f(xt;t;t)+TXt=1g(xt1;xt)+TXt=1h(xt)+TXt=1(vt) (2{15) Similarly,constraints( 2{8 )and( 2{13 )wouldbereplacedwith: 33

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2{17 )and( 2{18 ),respectively,oncew(xt;t;t)isknown.Theonlymodicationofthenetworkistoincludethecostassociatedwithviolatingtheperformanceconstraint. Whenevaluatingahospital,werecognizethattheaveragewaitingtimetobeassignedabedorhavingexpenseswithinbudgetarenottheonlymetricstoassessfacilityperformance.Indeed,itmaybenecessarytoincludemeasuresforfacilityutilization,likelihoodofpatientdiversion,andthelike.Regardless,wenotethatmoreperformanceconstraintscaneasilybeaddedtotheformulationsfortheBCP,RBCP,andRBCPwSproblems.Anincreaseinthenumberofperformanceconstraintsdoesnotincreasethetimetoobtainthesolutionsignicantly,asthereisonlyaneedtotaketheseadditionalconstraintsintoaccountinsettingupthenetworkandassigningalargearccostincaseanyoftheconstraintsareviolated.Therefore,ourmodelingapproachisrobustandadditionalconstraintscanbeconsideredwithoutincreasingthecomplexityoftheformulationsignicantly. 34

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Ourmodelisbasedonagenericviewofahospitalwherewehaveassumedthatthedemand(i.e.,patientarrivals)andservice(i.e.,beds)componentsarehomogeneous.Fromanaggregateplanningperspective,suchuniformitymaybeacceptable.However,inordertoapplythisresearchtooperationaldecisionsupportforhealthcaredelivery,thereareadditionalavenuesofresearchworthpursuing.First,ifcostdependsonallpreviousstages,forexample,thecostofmaintainingthebedsdependsnotonlyonthenumberofbedsbutalsothedurationofthebedsareinthesystem,thenthenumberofverticesinthenetworkwillbeexponentialwithrespecttoTandtheoptimalsolutiontothenetworkwillnotbesolvedwithapolynomialtimealgorithm.Consequently,alternativemodelformulationsandsolutiontechniquestodeterminetheoptimalbedplanwouldbenecessary.Second,recognizingthathospitalbedsarenotidentical,facilitycapacitycouldbeseparatedtodistinguishthevariousspecialties,withspecialty-specicdemandrates,lengthsofstay,andcosts.Indeterminingtheaveragewaitingtimeassociatedwithbeingassignedabed,wehaveusedclosed-formapproximationstocalculatethisstatistic.Therefore,weareimplicitlyassumingthatthisgeneraldistributionaccountsfordierenttypesofpatientsthatrequiredierenttypesofhospital-basedhealthcare.Thismaynotnecessarilybethecase,andshouldbeinvestigatedfurther.Third,ourworkcanbeexpandedtoincludemultipletypesofpatients(e.g.,electives,admissionscomingthroughtheemergencydepartment,andreferralsfromphysicians).Also,inestimatingthecostofpatientwaiting,weassumethatthiscostisidenticalregardlessofpatienttype.Clearly,forexample,thereshouldbedierentwaitingcostsassociatedplacinganemergencydepartmentadmissioninanappropriateunitversusaninappropriateunit.Assuch,representationsofpatientwaitingcostneedtobedevelopedinthepresenceofcongested,heterogeneousresources.Fourth,thecurrentformofourmodeldoesnotaccountforthepotentialtimedelaythatmayexistbetweenthedecisiontoexpand 35

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3 ]andtheongoingdebateregardingnurse-to-patientratios[ 4 ],theabilitytousephysicalcapacityhingesupontheavailabilityofsuitablemedicalpersonnel.Anaturalextensionofourmodelwouldbetoincorporateworkforceplanningtosimultaneouslydeterminethequantityandcompositionofthehealthcareresourcestoconstructacomprehensivecapacityplan. 36

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102 ],thehealthcareindustryaccountsforthelargestsectoroftheeconomyintheUnitedStates(US).Despiteadvancesinmedicaltechnologyand,thereby,theincreasinguseofmedicaldiagnostic,monitoring,andtreatmentequipment,thehealthcareindustryishighlylabor-intensive.AccordingtotheUSDepartmentofLabor,thehealthcareindustryprovided13.5millionjobsin2004,outofwhich13.1millionjobsareforwageandsalaryworkersandabout411,000arefortheself-employed[ 107 ].Itfollowsthatpersonnelwagesandsalariesaccountforthelargestportionofthetotalexpendituresforanyhealthcarefacility.Forinstance,hospitalsspendonaverageabout54percentofallexpendituresonwagesandsalaries[ 86 ].Hence,healthcarepersonnelplanning,i.e.,determiningtheappropriatemixofhealthcarepersonnel,neededtoprovidesafe,eective,timely,andcost-ecientservicetopatients[ 55 ],isanimportantproblem. Inpractice,bothininpatientandoutpatientfacilities,ahealthcareteam,comprisedofagroupofhealthcarepersonnelwithdierent,andcomplementary,skillsets,provideshealthcareservicestoindividualpatients[ 34 ].Themembersoftheteam(e.g.physicians,physician'sassistants,andregisterednurses)areresponsibletoperformasetoftasksrequiredforthediagnosis,monitoring,andtreatmentofthepatients.Someadditionaltasksmayhavetobeperformedbyotherpersonnel(e.g.,laboratorytechnicians,radiologicaltechnicians,andradiologists)whoarenotapartofthehealthcareteambutprovideassistancefordiagnosisandtreatment.Inthedeliveryofservicesbyhealthcareteams,thesafetyandeectivenessoftheserviceisensuredbytheappropriateselectionoftheservicecapabilityoftheteam,whereasthetimelinessandcost-eectivenessoftheserviceisensuredbytheappropriateselectionoftheservicecapacityoftheteam.Theservicecapabilityofateamischaracterizedbythecollectionoftheskillspossessed 37

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Inahealthcarefacility,therearetypicallymultipletypesofhealthcareteamseachwithdierentcapabilities.Thepatientsthatarrivetothefacilityareclassiedaccordingtotheirconditions(i.e.,acuitylevels)ormedicalrequests.Basedonthisclassication,eachpatientisassignedtoahealthcareteamandadmittedtoanexamination/treatment(E/T)roomthathastheequipmentnecessarytoprovidetheserviceneededbythepatient.AsthesetoftasksaresharedamongthemembersoftheteamandatypicalfacilityhasmultipleE/Trooms,ahealthcareteamusuallyservesmultiplepatientssimultaneously.Forinstance,whiletheteamwaitsforthetestresultsfromthelabforapatient,aregisterednursemaybecollectingspecimensfromanother,aphysician'sassistantmaybesuturingawoundofanother,andaphysicianaccompaniedbyaregisterednursemaybediscussingatreatmentplanwithanother.Inourwork,weconsidertwoparticularsettingswherehealthcareservicesareprovidedbyteams. ShandsatAlachuaGeneralHospitalinGainesville,Floridaisacommunityhospitalthatprovidesemergencymedicineservices.triagenurse,whodeterminestheacuitylevelofthepatientandidentieswhetherthepatientrequiresimmediate(i.e.,emergency)ordelayed(i.e.,urgent)care.Theemergencycareservicesaredeliveredbyanemergencycare(EC)team,whichiscomposedofphysiciansandregisterednurses,andtheurgentcareservicesaredeliveredbyanurgentcare(UC)team,whichisalsocomposedofphysiciansandregisterednurses.Wenotethatthetriagenursedoesnotbelongtoeitheroftheseteams,butactsasagatekeepertorouteanarrivingpatienttoeitheroftheteams.ThereareeightandtwoE/TroomsdedicatedtotheECandUCteams,respectively.Moreover,theECteamhasthecapabilitytoattendtourgentcarepatients,buttheUCteamdoesnothavethecapabilitytoattendtoemergencycarepatients.Finally,emergencycarepatientshavepreemptivepriorityoverurgentcarepatients.Thatis,ifanemergencycarepatientarrivestotheED,whileallE/TroomsdedicatedtotheEC 38

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TheWomen'sClinicattheStudentHealthCareCenterattheUniversityofFloridainGainesville,Floridaprovideswomen'shealthcareservices.Theoutpatientclinic(OC)servesnotonlynon-acutepatients,i.e.,thosewhoneedroutineservices,buttreatsacutepatientsalso.Theservicesforthediagnosisandtreatmentofacuteillnessesandabnormalitiesareprovidedbyaphysician(P)team,whichiscomposedofaphysicianandaphysician'sassistant.Theroutineclinicalservicesaredeliveredbyanursepractitioner(NP)team,whichiscomposedofanursepractitionerandaregisterednurse.TherearefourE/TroomsdedicatedtotheNPteamandtworoomstothePteam.Moreover,thePteamhasthecapabilitytoattendtonon-acutepatientsbuttheNPteamdoesnothavethecapabilitytoattendtoacutepatients.Finally,acutepatientshavenon-preemptivepriorityoverthenon-acutepatients.Thatis,ifanacutepatientarrivestotheOC,whileallE/TroomsdedicatedtothePteamareoccupiedbyotherpatientsandoneofthemisanon-acutepatient,thenthePteamdoesnotinterruptservicetothenon-acutepatientandtheacutepatienthastowaituntilanE/TroomdedicatedtothePteambecomesavailable. Inthesettingswediscussedabove,theservicecapabilitiesofthehealthcareteamsarexedasdictatedbytheserviceneedsofthepatients,andpersonnelplanningismainlyconcernedwithdeterminingtheservicecapacityoftheteams.Indeterminingtheservicecapacity,however,administratorsmusttakeseveraladditionalfacilitycapacity,budgetary,andlegislativeconstraintsintoaccountthatlimittheminimumandmaximumtotalnumberofeachpersonneltypeemployed.Forinstance,budgetconstraintsmaylimitthetotalnumberofphysiciansemployed,whereaslegislatednurse-to-patientstangratiosmayprescrobealowerlimitonthetotalnumberofregisterednurses.Therefore,giventhelowerandupperboundsonthetotalnumberofpersonnelwithdierentskillsthatcanbe 39

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Inthispaper,motivatedbythepracticalsettingswediscussedabove,weaddressthelong-termhealthcareteamservicecapacityplanning(HCTSCP)probleminthecontextofhealthcarefacilitieswheretherearetwopatientclassesandtwotypesofteams.Weassumethatthecapabilityofeachtypeofteamisknown,andthenumberofE/Troomsallocatedtoeachtypeofteamisxedovertheplanninghorizon.Therefore,theservicecapacityofateamcanonlybechangedbymodifyingthecongurationoftheteam.Moreover,theservicecapacityofateamcanbequantiedbytheservicerateoftheE/Troomsdedicatedtotheteam,whichisthereciprocaloftheaveragetimethatthepatientsspendintheE/Troomspriortodischargeortransfertoanotherdepartment.Weformulateanon-linearbinaryintegerprogrammingmodeltodeterminetheservicecapacityplanforthehealthcareteamssuchthathealthcareservicesaredeliveredinatimely(i.e.,theaveragetimeapatientfromaparticularclassspendsinthesystemdoesnotexceedapre-speciedthreshold)andcost-eective(i.e.,thetotalcostsassociatedwithchangingservicecapacitybyhiringadditionalpersonnel,reassigningexistingpersonnelorlayingoexistingpersonnelandoperatingtheservicecapacityareminimized)manneroveragivenplanninghorizonconsideringsomeadditionalconstraints. Toestimatetheaveragetimeapatientfromaparticularclassspendsinthesystem,wedevelopqueuingmodelsanddecompositionbasedapproximation(DBA)methods. 40

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Theremainderofthispaperisorganizedasfollows.InSection 3.2 ,wereviewtherelatedliterature.Section 3.3 presentsourcapacityplanningmodelforHCTSCPandshowhowitcanbeinterpretedasanetworkowmodel.WedevelopqueueingmodelsfortheEDandOCsettingsandpresentDBAmethodstoanalyzethesemodelsinSection 3.4 .InSection 3.5 ,wepresentresultsfromourcomputationalstudythatevaluatestheaccuracyoftheapproximationsfromSection 3.4 andinvestigatethecomputationalperformanceoftheformulationpresentedinSection 3.3 .Section 3.6 includesadiscussionoftheresultsandsuggestsfutureresearchdirections. 41

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61 ].Burkeetal.[ 28 ]provideanexcellentreviewoftheexistingworkinthisarea. Analyticalworkthatfocusesonmedium-tolong-termpersonnelplanningisrelativelylimitedinscope.SchneiderandKilpatrick[ 97 ]developoptimizationmodelsforpersonnelplanninginhealthcarefacilities.KaoandTung[ 62 ]presentalinearprogrammingmodelfortheaggregate(nursing)workforceplanningproblem,whichislaterextendedbyBruscoandShowalter[ 27 ]toaccountfortheexogenousimpactofnursingshortage.KroppandCarlson[ 72 ]proposetheintegrateduseofoptimizationandsimulationmodeling.ExistingworkthatusessimulationmodelingisreviewedinJunetal.[ 58 ].Althoughtheearlierworkinthisareaprimarilyfocusesontheperspectiveofthehealthcareproviderbyplacinganemphasisonminimizingthecostofpersonnelresourcesormaximizingtheutilizationofpersonnelresources,thereisaneedtotakethepatient'sperspectiveintoaccountbyconsideringthetimeliness(e.g.,minimizingofthewaitingtimeand/ortimeinsystem)ortheavailability(e.g.,minimizingtheprobabilityofndingallserversbusyandbeingdivertedtoanotherserviceprovider).Tothisend,wefocusonminimizingthesumofcapacitycostsandcostofnotprovidingserviceinatimelymanner,whileensuringthattheaveragetimeinsystemforapatientdoesnotexceedapre-speciedthreshold.Tothisend,wedividetheplanninghorizonintodiscretetimeperiodsofequallengthandassumethatthesystemachievessteadystateineachoftheseintervals.Thisallowsustousequeueinganalysistocapturethestochasticbehaviorofthesystemandcomputethe 42

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5 ]. Intheliterature,thereisanumberofstudiesthatinvestigatequeueingsystemsthatarecloselyrelatedtothesettingweconsider.StanfordandGrassmann[ 105 ]derivetheexpectedwaitingtimeinacallcenterwithunilingualandbilingualserversservingmajority-andminority-language-usecustomers.Aminority-language-usecustomercanonlybeservedbyabilingualserver,andthetypeofacustomerisnotbeknownpriortotherstservice.Theserviceratesforallservertypesarethesame,i.e.,independentofthecustomertype.Green[ 44 ]andShumsky[ 99 ]alsoinvestigateexpectedwaitingtimeofasystemwithtwotypesofcustomersandlimited-andgeneral-useservers.Bothofthemconsiderthecasewheretheserviceratesdependonthetypeoftheserver.Hence,thedistinguishingcharacteristicofthesystemweinvestigateisthedependenceoftheservicetimeoncustomerclassforoneoftheservertypes.Thisseeminglyasimpleattributeofthesystemleadstosignicantmodelingchallenges. Inqueueingtheory,toobtainthestationaryprobabilitiesofasystem,themajorapproachesusedincludethematrixanalyticmethods[ 44 60 74 75 87 105 ],power-seriesalgorithm(PSA)approach[ 19 20 54 71 ],andDBAmethod[ 99 ].ThematrixanalyticmethodsformulatethesystemasaMarkovchaintowhichthestationaryprobabilityhasthematrix-geometricformn=0Rn,whereratematrixRcanbeobtainedthroughaniterativealgorithm[ 87 ].ThePSAapproachrstrepresentsthestationaryprobabilitiesofasystemaspower-seriesexpansionsofthetracintensityofthesystem,andthenrecursivelysolvesforthecoecientsofthepower-seriesexpansionsbyusingthesetofstationaryequations.KaoandWilson[ 64 ]comparetheperformanceofthePSAandmatrixanalyticmethodswiththreeiterativealgorithmsproposedin[ 60 74 75 ],andconcludethatthePSAperformsextremelywellintermsofcomputationalspeed,thoughitmayencounterdicultiesinparametersettingswhichmayleadtolossesinaccuracy.Shumsky[ 99 ]proposestheDBAmethod.TheDBAmethodrstdividesthestatespace 43

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99 ]illustratesthatthismethodgeneratesperformancemeasuresrapidlywithsucientaccuracythatcanbeusedincallcentercapacitydecisions.Inourqueueinganalysis,wealsousethisDBAmethod. Therearetwotypesofhealthcareteams,indexedbyj=1;2,andtwotypesofE/Trooms.Teamtype1canprovideservicetobothpatientclasseswithdierentserviceratesinitsdedicatedE/Trooms,type1,whileteamtype2canonlyprovideservicetopatientclass2initsdedicatedE/Trooms,type2.LetrjdenotethenumberofE/Troomsallocatedtoteamtypejforj=1;2,andrbetheassociatedtwo-dimensionalvectorrepresentingtheroomallocation.Aswediscussedearlier,theservicecapacityofeachtypeofteamcanonlybechangedbymodifyingthecongurationoftheteamandcanbequantiedbytheserviceratesoftheassociatedE/Trooms.Giventhelowerandupperboundsonthenumberofpersonnelwithdierentskillsthatcanbeemployedinthefacilityandthetypesofpersonnelthatmustbeincludedinaparticulartypeofteam,letKjdenotethetotalnumberofpossiblecongurationsforteamtypej,indexedbyk=1;:::;Kj.Supposethatitispossiblefortheadministratorstoestimatethe 44

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Wehavetwosetsofdecisionvariables.Letxjkttakethevalueofoneifforteamtypejcongurationkisselectedinperiodtforj=1;2,k=1;:::;Kjandt=1;:::;T,andzerootherwise.Also,letxjtbeaKj-dimensionalvectorassociatedwiththeteamcongurationdecisionforteamtypejinperiodt.Finally,letijtdenotetheservicerateforclassipatientswhoaretreatedintypejE/Troomsinperiodtfori=1;2,j=1;2andt=1;:::;T,andtbea22matrixassociatedwiththeserviceratesoftheE/Troomsinperiodt. Weletwi(t;t;b;r)andsi(t;t;b;r)representthewaitingcostofclassipatientsandtheaveragetimespentinthesystembyclassipatients,respectively,asafunctionoftheservicerateoftheE/Troomstinperiodt,patientarrivalratestinperiodt,maximumnumberofpatientsallowedinthefacilitybandE/Troomallocationr.Also,weletcj(xjt1;xjt)denotethecostofmodifyingthecongurationoftypejteamfromperiodt1totandoj(xjt)denotethecostofemployingthepersonnelnecessaryforthechosencongurationforteamtypejinperiodt.Assumingthatallacquiredadditionalpersonnelcapacityisavailableandbecomeseectiveinthesameperiod,theHCTSCPproblemcanbeformulatedasanon-linearbinaryintegerprogrammingproblemasfollows: min2Xi=1TXt=1wi(t;t;b;r)+2Xj=1TXt=1cj(xjt1;xjt)+2Xj=1TXt=1oj(xjt) (3{1)subjecttoKjXk=1xjkt=18j;t 45

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Theobjectivefunction( 3{1 )minimizesthesumofthecostassociatedwiththetimethatthepatientsspendinthesystem,thecostofmodifyingteamcongurationstochangetheservicecapacityoftheteams,i.e.,serviceratesoftheassociatedE/Trooms,andthecostofemployingthepersonnelnecessaryfortheselectedteamconguration.Constraints( 3{2 )stipulatethatonecongurationmustbeselectedforeachteamtypeineachplanningperiod.Constraints( 3{3 )assigntheserviceratesforE/Troomsaccordingtotheselectedcongurationsforeachteamtypewithrespecttodierentpatientclasses.Notethat12t=0fort=1;:::;Tdueto12k=0fork=1;:::;K2.Constraints( 3{4 )imposeupperlimitsontheaveragetimethatpatientsspendinthesystemforeachpatientclass.Constraints( 3{5 )limitstheamountoffundsthatcanbeallocatedtochangingteamcongurationineachplanningperiod.Finally,constraints( 3{6 )and( 3{7 )ensuretheintegralityofteamcongurationandnon-negativityofserviceratedecisionvariables,respectively. HCTSCPisadicultnon-linearbinaryintegerprogrammingproblemwithnon-linearconstraints.Notethat,however,HCTSCPcanberepresentedbyaT+1-partitenetwork,whereeachlayerinthenetworkrepresentsatimeperiodt=0;:::;Tintheplanninghorizon.Let(k1;k2;t)denotethefacilitywhencongurationkjfortypejteamisusedinperiodt.Layert=0includeasinglenode(k1;k2;0),whichdenotestheinitialcongurationsfortype1andtype2teams.AsupercialsourcenodeSisconnectedtonode(k1;k2;0)onlywithzeroarccost.Eachlayert=1;:::;TcontainsK1K2nodes,eachofwhichrepresentsafeasiblepairofteamcongurationsforthetwoteamsinperiodt.Thecostofthearcsconnectinganode(k1;k2;t1)inperiodt1toanode(k1';k2';t)inperiodtisgivenbyP2i=1wi(t;t;b;r)+P2j=1cj(xjt1;xjt)+P2j=1oj(xjt),wherexjkj't=1forj=1;2,fort=1;:::;T.However,foragivennode(k1;k2;t),if 46

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3{4 )orconstraints( 3{5 )isviolated,thenthecostoftheincomingarcstothisnodearesettoM,whereMisaverylargenumber.Finally,eachnodeinlayert=TisconnectedtoasupercialsinknodeDonlywithzeroarccost.Figure 3-1 providesanexampleofthenetworkrepresentationforHCTSCPforK1=3,K2=3,andT=2.Foreachteamtype(a1;a2)representsacongurationwherethenumberoftype`personnelintheteamisa`for`=1;2.Inthisgure,apathfromthesupercialsourcenodetoSthesupercialsinknodeDrepresentsateamcapacityplanovertheplanninghorizon.TheHCTSCPproblemndsacapacityplanwithminimumcost,iftheshortestpathonthisgraphdoesnotcontainanyarcwithcostM.Otherwise,theproblemisinfeasible.Inthenextsection,weexplainhowweobtaintheaveragetimeinsystemforeachpatientclass. Figure3-1. AnillustrationofthenetworkrepresentationforHCTSCP 47

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Thetransitionrateoutofstate(n1;n2;m)2Spisthesumofarrivalanddepartureratesofbothpatientclasses,thatis,1I(n1
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Thepossibletransitionsenterstate(n1;n2;m)fortheEDsetting EventTransitionprobability 1)Aclass1patientarrivestothesystem 2)Aclass1patientarrivestothesystemandpreemptsaclass2patientservedinatype1E/Troom 3)Aclass2patientarrivestothesystem,andisadmittedimmediatelytoatype2E/Troomorjoinsthequeueofclass2patients 4)Aclass2patientarrivestothesystem,andisadmittedimmediatelytoatype1E/Troom 5)Aclass-1-patientdepartureresultsineitherfreeingatype1E/Troomorstartingtheserviceofawaitingclass1patient [(n1+1)^r1]11P(n1+1;n2;m)I(n10)I(n1+m=r1)I(n2mr2) 7)Aclass-2patientdeparturefromatype1E/Troomresultsinstartingtheserviceofawaitingclass2patient 8)Aclass-2-patientdeparturefromatype2E/Troomresultsinstartingtheserviceofawaitingclass2patient [(n2+1m)^r2)]22P(n1;n2+1;m)I(n2
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50

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Thetransitionrateoutofstate(n1;n2;m)2Snpisthesumofarrivalanddepartureratesofbothpatientclasses,thatis,1I(n1
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Two-dimensionalCTMCapproximation twoindependentsimplesubsystemsandusetheperformancemeasuresofthesesimpliedsubsystemstoadjusttheresultsobtainedinstep(3). 3-2 .Subsystem1containsthetype1team,type1E/Troomsandthepatientsthatareservedinthetype1E/Trooms.Similarly,subsystem2containsthetype2team,type2E/Troomsandthepatientsthatareservedinthetype2E/Trooms.Wethenuseatwo-dimensionalCTMCwithstate(z;n2')toapproximatetheoriginalthree-dimensionalCTMC,wherezisthetotalnumberofpatientsinsubsystem1andn2'isthenumberofclass2patientsinsubsystem2.Notethatzincludesthenumberofclass1andclass2patientsinsubsystem1,andn2'includesthenumberofclass2patientsservedinsubsystem2only.Thestatespaceofthetwo-dimensionalCTMCisdenedbyS'np=f(z;n2'):0zr1+b1;0n2'r2I(z
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wherePc2=[2P(N2'=r2jZr1)]=[1+2P(N2'=r2jZr1)]:NotethattheprobabilityP(N2'=r2jZr1)canbeapproximatedbyP(N2'=r2jZ
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z=(r1E[MjZ=z])11+E[MjZ=z]21; wherethevalueofE[MjZ=z]followsfromequation( 3{9 ). 3{10 )andPon1istheprobabilitythatatype1serverisavailableforawaitingclass2patient,andcanbeapproximatedbyPon1=P(Z=r1jN2'>r2)(r1+2)=(r1+2+1). z=(min(z;r1)E[MjZ=z])11+E[MjZ=z]21; wherethevalueofE[MjZ=z]iscomputedfromequation( 3{8 )for0zr1,andfromequation( 3{9 )forr1
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1P(N2'>r2jZr1)P(Z
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Notethattherstterminequation( 3{15 )representstheprobabilitythatanarrivingclass2patientndsalltype2E/Troomsoccupiedandoverowstosubsystem1immediately.Thesecondtermrepresentstheprobabilitythatanarrivingclass2patientwaitsinsubsystem2untilatype1E/Troombecomesavailable.LetE[ 3{12 )and( 3{13 )aredecreasedproportionally,i.e., 3{12 )and( 3{13 )areincreasedproportionally. 56

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106 ].AGEalgorithmavoidsunnecessaryrowoperationsbyconsideringthefactthatQisabandedmatrix,containingalargenumberofzeroelements.TheDBA,GE,andAGEmethodsareimplementedusingC++programminglanguage,andthenumericalresultsreportedareobtainedusingapersonalcomputerwitha3.0GHzPentiumIVprocessorand1GBRAMmemory. OurparameterchoicesforourcomputationalstudyarebasedonthedatacollectedfromaparticipatingED.Inthebasecase,thereareeightandtworoomsallocatedtotype1andtype2teams,respectively,i.e.,(r1;r2)=(8,2);serviceratesforclass1andclass2patientsintype1E/Troomare0.25and0.80patients/hour,respectively,i.e.,(11;21)=(0.25,0.80);servicerateforclass2patientsintype2E/Troomare0.75patients/hour,i.e.,22=0.75;andthemaximumnumberofpatientsfromeachclassallowedinsystemare20and20,respectively,i.e.,(b1;b2)=(20,20).Wenotethatinpractice,anEDisrequiredbylawtoadmitallthepatientsthatrequestemergencymedicineservices.Therefore,essentially,thewaitingroomcapacityisinnite,andnoarrivingpatientisdeniedofentrytothesystembecauseoflackofwaitingroomcapacity.However,whenthereisnotenoughservicecapacity,thenanarrivingpatientcanbedivertedtoasisterhospital.Therefore,formodelingpurposes,weincludealimitonthenumberofpatientsofeachtypeinthesystem. 57

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InTable 3-2 ,wereportthesizeoftheCTMCaswellastheCPUtime(inseconds)requiredtoobtainthesolutionusingtheDBA,AGE,andGEmethods.Inthepreemptivecase,thesizeoftheCTMCmodelgrowsifr1orr2increases.Inaddition,anincreaseinr1hasamoresignicanteectthanthatinr2.Similarbehaviorcanbeobservedinthenon-preemptivecase,however,theimpactofincreasingr1onproblemsizeinnon-preemptivecaseismoresignicantthanthatinthepreemptivecase.Furthermore,underthesameE/Troomallocation,theproblemsizeofnon-preemptivecaseisatleasttwicelargerthanthatofpreemptivecaseforthetestedscenariosinTable 3-2 .OurresultsshowthatthetimetoobtaintheapproximateresultsusingtheDBAmethodisnegligibleandAGEisconsiderablymoreeectivethanGEinobtainingtheexactsolution. Table3-2. ComputationalrequirementoftheDBA,AGE,andGEmethods PreemptiveNon-Preemptive 224470.00.00.527070.00.310.8244530.00.00.627130.00.310.9284650.00.00.627250.00.311.2 424610.00.00.642250.00.736.6444810.00.00.742450.00.737.9485210.00.00.942850.00.739.8 825130.00.00.866090.01.9116.1825850.00.01.266810.02.0129.2827290.00.02.368250.02.1161.0 Unitofrunningtime:Second 58

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3-3 and 3-4 ,wereportthepercentageerrorassociatedwiththeexpectedtimeinsystemforthetwopatientclassesobtainedbytheDBAmethod(whencomparedtotheexactsolutionobtainedbytheAGEmethod)inthepreemptiveandnon-preemptivecase,respectively.Table 3-3 showsclearlythattheaveragetimeinsystemforclass1patientscanbecorrectlycomputedbytheDBAmethodbecausesubmodel1inSection 3.4.1 capturestheexactbehaviorofclass1patientsintheoriginalCTMC.Forclass2patients,therstexperimentshowsthatfortheinstanceswiththesamelevelsofr1and,theabsolutepercentageerror(APE)tendstodecreaseasr2increases.Forexample,fortheinstanceswithr1=2and=0.6,APEdecreasesasr2increases.Inaddition,forthesamelevelsofr1/r2and,APEtendstodecreaseasr1orr2increasesassystemutilizationislow,suchastheinstanceswithr1/r2=1and=0.6,APEdecreasesasr1(orr2)increases.Inotherwords,foragivensettingofr1orr1/r2and,DBAperformsbetterintheproblemswithlargerr2.Inthesecondexperiment,9scenariosofservicecapacityaretested.Forthesamelevelof(11;21),APEtendstoincreaseas22increasesfortheinstanceswith=0.6,whichisoppositetotheresultsoftheinstanceswith=0.9,whereAPEtendstodecreaseas22increases.Last,bothexperimentsshowthatDBAtendstounderestimateclass2patients'expectedtimeinthesystemwhenthesystemutilizationislow,i.e.,=0.6,whileoverestimateclass2patients'expectedtimeinthesystemwhenthesystemutilizationishigh,i.e.,=0.9. Table 3-4 showstheresultsfornon-preemptivecase.Intherstexperiment,weobservethatfortheinstanceswiththesamelevelsofr1and,theAPEforclass2patientstendstodecreaseasr2increases,whichisthesameasthepreemptivecase.Inthesecondexperiment,forthesamelevelsof(11;21)and,APEtendstoincreaseas22increasesforbothclass1andclass2patients.Inaddition,DBAtendstounderestimateclass1patients'expectedtimeinsystemwhileittendstooverestimateclass2patients'expectedtimeinsystem.Forclass1patients,APEislessthan5percentforalltestedinstances,i.e.,DBAyieldsamorereliableestimateforclass1patients'timeinsystem. 59

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Relativeandpercentageerror:preemptivecase 220.0(0.0%)-5.3(-6.1%)0.0(0.0%)-5.3(-4.4%)0.0(0.0%)-0.1(0.0%)0.0(0.0%)10.1(3.1%)40.0(0.0%)-5.2(-3.4%)0.0(0.0%)-8.2(-4.1%)0.0(0.0%)-10.0(-3.6%)0.0(0.0%)-9.5(-2.3%)80.0(0.0%)-3.1(-1.1%)0.0(0.0%)-6.6(-1.8%)0.0(0.0%)-10.8(-2.4%)0.0(0.0%)-12.5(-2.2%) 420.0(0.0%)-3.2(-4.0%)0.0(0.0%)-1.2(-1.1%)0.0(0.0%)9.1(5.6%)0.0(0.0%)29.9(10.6%)40.0(0.0%)-3.8(-2.5%)0.0(0.0%)-5.1(-2.7%)0.0(0.0%)-3.6(-1.4%)0.0(0.0%)1.3(0.3%)80.0(0.0%)-2.5(-0.9%)0.0(0.0%)-5.3(-1.5%)0.0(0.0%)-8.5(-2.0%)0.0(0.0%)-10.1(-1.8%) 820.0(0.0%)-2.3(-3.1%)0.0(0.0%)0.6(0.6%)0.0(0.0%)14.9(11.0%)0.0(0.0%)51.5(22.9%)40.0(0.0%)-2.7(-1.9%)0.0(0.0%)-2.9(-1.6%)0.0(0.0%)2.2(0.9%)0.0(0.0%)15.4(4.5%)80.0(0.0%)-1.9(-0.7%)0.0(0.0%)-3.9(-1.1%)0.0(0.0%)-5.6(-1.3%)0.0(0.0%)-5.9(-1.1%) Average0.0(0.0%)3.3(2.6%)0.0(0.0%)4.4(2.1%)0.0(0.0%)7.2(3.1%)0.0(0.0%)16.2(5.4%) Experiment2:Eectsofservicecapacity,(r1;r2)=(8,2) (0.20,0.70)0.650.0(0.0%)-2.4(-3.2%)0.0(0.0%)0.3(0.4%)0.0(0.0%)14.1(10.3%)0.0(0.0%)49.3(21.7%)0.750.0(0.0%)-3.6(-4.7%)0.0(0.0%)-1.6(-1.6%)0.0(0.0%)10.5(7.3%)0.0(0.0%)41.4(17.5%)0.850.0(0.0%)-4.9(-6.2%)0.0(0.0%)-3.5(-3.4%)0.0(0.0%)7.1(4.8%)0.0(0.0%)34.6(14.1%) (0.25,0.80)0.650.0(0.0%)-1.2(-1.7%)0.0(0.0%)2.3(2.5%)0.0(0.0%)18.4(14.1%)0.0(0.0%)59.4(27.6%)0.750.0(0.0%)-2.3(-3.1%)0.0(0.0%)0.6(0.6%)0.0(0.0%)14.9(11.0%)0.0(0.0%)51.5(22.9%)0.850.0(0.0%)-3.4(-4.4%)0.0(0.0%)-1.1(-1.1%)0.0(0.0%)11.7(8.3%)0.0(0.0%)44.5(19.0%) (0.30,0.90)0.650.0(0.0%)-0.4(-0.6%)0.0(0.0%)3.9(4.3%)0.0(0.0%)22.0(17.5%)0.0(0.0%)68.0(33.2%)0.750.0(0.0%)-1.3(-1.8%)0.0(0.0%)2.3(2.5%)0.0(0.0%)18.6(14.2%)0.0(0.0%)60.0(27.9%)0.850.0(0.0%)-2.3(-3.0%)0.0(0.0%)0.8(0.8%)0.0(0.0%)15.5(11.4%)0.0(0.0%)53.0(23.7%) Average0.0(0.0%)2.4(3.2%)0.0(0.0%)1.8(1.9%)0.0(0.0%)14.8(11.0%)0.0(0.0%)51.3(23.1%) Unit:min.(%)

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Relativeandpercentageerror:non-preemptivecase 22-3.2(-2.8%)1.1(1.3%)-5.6(-3.3%)-4.5(-3.7%)-6.2(-2.3%)-14.8(-8.1%)13.7(3.3%)-23.6(-7.4%)4-3.1(-2.7%)2.3(1.5%)-5.9(-3.5%)-2.6(-1.3%)-9.1(-3.5%)-10.4(-3.8%)9.9(2.4%)-14.2(-3.4%)8-2.3(-2.0%)3.8(1.3%)-5.2(-3.1%)0.2(0.1%)-9.4(-3.6%)-4.5(-1.0%)5.9(1.4%)-4.6(-0.8%) 42-2.7(-1.6%)9.5(12.0%)-5.5(-2.4%)4.1(3.9%)-9.5(-2.9%)-10.9(-7.0%)-12.9(-2.7%)-32.6(-12.0%)4-2.8(-1.6%)10.8(7.2%)-6.1(-2.6%)6.0(3.2%)-10.9(-3.3%)-8.8(-3.5%)-15.0(-3.1%)-27.0(-7.1%)8-2.1(-1.2%)9.1(3.1%)-5.6(-2.4%)9.1(2.6%)-11.2(-3.4%)-1.5(-0.3%)-16.0(-3.3%)-11.1(-2.0%) 82-1.6(-0.5%)19.6(26.6%)-4.2(-1.1%)19.0(20.4%)-8.3(-1.7%)8.5(6.6%)-12.4(-2.0%)-13.0(-6.1%)4-1.7(-0.6%)21.4(14.7%)-4.8(-1.3%)22.5(12.7%)-10.0(-2.1%)11.0(4.8%)-15.2(-2.5%)-13.7(-4.1%)8-1.4(-0.5%)11.5(4.0%)-4.7(-1.3%)25.2(7.3%)-10.8(-2.3%)17.5(4.2%)-17.0(-2.8%)-1.4(-0.3%) Average2.3(1.5%)9.9(8.0%)5.3(2.3%)10.3(6.1%)9.5(2.8%)9.8(4.4%)13.1(2.6%)15.7(4.8%) Experiment2:Eectsofservicecapacity,(r1;r2)=(8,2) (0.20,0.70)0.65-1.5(-0.5%)20.7(28.1%)-3.9(-1.0%)19.8(21.2%)-7.7(-1.6%)7.4(5.7%)-11.5(-1.9%)-19.5(-9.1%)0.75-1.8(-0.6%)22.7(29.7%)-4.4(-1.2%)21.7(22.3%)-8.5(-1.8%)8.1(5.9%)-12.5(-2.1%)-22.5(-10.0%)0.85-2.0(-0.6%)24.4(31.0%)-4.8(-1.3%)23.4(23.2%)-9.3(-2.0%)8.8(6.2%)-13.5(-2.2%)-25.1(-10.8%) (0.25,0.80)0.65-1.4(-0.5%)17.8(24.9%)-3.7(-1.0%)17.2(19.2%)-7.4(-1.6%)7.9(6.4%)-11.3(-1.9%)-10.6(-5.3%)0.75-1.6(-0.5%)19.6(26.6%)-4.2(-1.1%)19.0(20.4%)-8.3(-1.7%)8.5(6.6%)-12.4(-2.0%)-13.0(-6.1%)0.85-1.8(-0.6%)21.3(28.0%)-4.6(-1.2%)20.6(21.4%)-9.1(-1.9%)9.1(6.8%)-13.4(-2.2%)-14.8(-6.7%) (0.30,0.90)0.65-1.4(-0.4%)15.4(22.1%)-3.5(-0.9%)15.1(17.4%)-7.1(-1.5%)7.9(6.7%)-11.0(-1.8%)-4.0(-2.1%)0.75-1.5(-0.5%)17.1(23.8%)-4.0(-1.1%)16.7(18.6%)-7.9(-1.7%)8.6(6.9%)-12.1(-2.0%)-6.2(-3.1%)0.85-1.7(-0.6%)18.7(25.3%)-4.4(-1.2%)18.3(19.6%)-8.7(-1.8%)9.1(7.1%)-13.1(-2.1%)-8.2(-3.9%) Average1.6(0.5%)19.8(26.6%)4.2(1.1%)19.1(20.4%)8.2(1.7%)8.4(6.5%)12.3(2.0%)13.8(6.4%) Unit:min.(%)

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Weconsiderthepersonnelplanningproblemoverathree-yearplanninghorizonwheretheunitplanningperiodcorrespondstoaquarterofayear,i.e.,T=12.Weassumethat,forallt,theinitialserviceratesforclass1andclass2patientsare(110;210;220)=(0.25,0.80,0.75)patient/hour,theallowablemaximumnumbersofpatientsinsystemare(b1;b2)=(20,20)patients,theE/Troomallocationis(r1;r2)=(8,2),andtheupperboundontheamountoftimethatapatientspendsinthesystemare(1;2)=(4.75,4.00)hours.Weconsideracasewheretherearetwotypesofpersonnelincludedineachteam,andtherearesixandfourfeasiblecongurationsfortype1andtype2teams,respectively.Table 3-5 showsthenumberofdierenttypesofpersonnelwithdierentskillsetsincludedineachcongurationandthecorrespondingservicecapacityaswellasoperatingcostforeachteam.Withoutlossofgenerality,weassumethatpatientwaitingcostsincreaselinearlywithpatients'timeinsystem,unitdelaycostforeachpatientclass(UP1t;UP2t)aresetto($400,$100)/hourperpatient,andthefundsthatcanbeallocatedtochangetheservicecapacityoftheteamsinthefacilityt=$17,000forallt.Finally,weassumethatpersonnelhiringorterminationcostsarezero. AnexaminationofemergencymedicinepracticesshowthatpatientarrivalstotheEDexhibitseasonality.Wegeneratethetotalquarterlyarrivalrateofpatientsusingaseasonallyadjustedtrendline,representedbyafunctionoftheformt=mod(t;4)u(0+bt)whereiisthequarterlyseasonalityfactorforseasoni(wherewehavethefollowingestimatesfortheseasonalityfactors1=0.8,2=1.0,3=1.2,and4=1.0),uis 62

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Teamcongurations Type1TeamType2Team 1240.230.7556,0001110.6011,5002250.250.8063,0002120.7516,5003260.280.8570,0003221.0023,0004340.351.0070,0004231.1028,0005350.401.1077,0006360.451.2084,000 Unitofij:patients/hour.Unitofcost:$/quarter auniformlydistributedrandomnumber(wherewehaveuU[0:8;1:2]),0=2,b=0.04,andt=1;:::;12.Notethatweimplicitlyassumethatthepatientdemandincreaseslinearlyby2percenteveryquarter.Weassumethatthefractionofclass1patientisfit=0:85forallt,i.e.,1t=f1ttand2t=(1f1t)t.WenotethatourparameterchoicesaremainlyaccordingtothecharacteristicsofthedatacollectedfromtheED,whichisrepresentedbythepreemptivecase.Inordertoeliminatetheeectsthatmaybeduetoaspecichealthcarefacility,weusethesamesetofparametersforthenon-preemptivecasealso. Inourstudy,weconsideredthreeexperimentalfactorsincludingthefractionofclass1patients,f1t,theunitpatienttreatmentcostforeachpatientclass,(UP1t;UP2t),andthemaximumallowableaveragetimeinsystemforeachpatientclass,(1;2).EachparameteristestedatthreelevelsaslistedinTable 3-6 .Foreachexperimentalsetting,wegenerated25randominstancesofthepatientarrivalstreamsovertheplanninghorizon.WesolvedalltheinstancesforeachofthesettingsbyusingtheHCTSCPmodelwiththeAGEmethodandwiththeDBAmethodtocomparetheperformanceofthetwomethods. Table3-6. Parametersettings ParametersLevel1Level2Level3 Experiment1f1t(%)808590 Experiment2UP1t($/hour/patient)200400800UP2t($/hour/patient)50100200 Experiment31(hours)4.504.755.002(hours)3.504.004.50 63

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3.3 ,webuildanetworktorepresenttheHCTSCPproblem.Tocomputethepatientdelaycostforeacharc,weobtainaveragetimeinsystemforeachpatientclassapproximately(exactly)usingtheDBA(AGE)method.WenotethatwedonotusetheGEmethodinthisexperiment,sincetheAGEmethodisshowntobeconsiderablymoreecientinSection 3.5.1 .Afterweconstructthenetwork,wendtheshortestpathusingDijkstra'salgorithm[ 2 ].Theresultsshowthatthenetworkforthe12-periodHCTSCPproblemwithsixandfourfeasiblecongurationsfortype1andtype2teams,respectively,contains290nodesand6,384arcs.Theaveragetimerequiredtoobtaintheoptimalcapacityplanis11.0secondsforpreemptivecaseand549.8secondsfornon-preemptivecaseifweusetheAGEmethodinbuildingthenetworkforaninstanceoftheHCTSCPproblem.Incontrast,theDBAmethodrequires0.03seconds,onaverage,forboththepreemptiveandnon-preemptivecases.Therefore,usingtheDBAmethodinbuildingthenetworkforaninstanceoftheHCTSCPproblemsisconsiderablymoreecientthanusingtheAGEmethod. InordertomeasuretheaccuracyoftheDBAmethod,wecomparetheteamcapacityplansgeneratedbythetwoapproachesbasedonasimilarityindex.Specically,wecountthenumberofperiodswhereteamcongurationschosenarethesameinbothapproachesandthendividethiscountingby12foreachcareteamtypetodeterminethevalueofthesimilarityindex.TheresultsofourcomparisonforeachexperimentalfactorsaresummarizedinTables 3-7 3-8 ,and 3-9 .Cellsrangefrom0percent(absolutelydierent)to100percent(perfectlysimilar).Forinstance,ifacellhasavalueof75percent,thenamong12periods,theHCTSCPmodelwithDBAgivesthesameresultsin8ofthemastheHSCTSCPmodelwithGE. InTable 3-7 ,weobservethattheHCTSCPmodelwithDBAworksverywellinthepreemptivecase.However,inthenon-preemptivecase,asthefractionofclass1patientsincreases,theperformanceoftheHCTSCPmodelwithDBAdeterioratesasitoverestimatestherequiredservicecapacityoftype2teambychoosingtheteam 64

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Impactofthefractionofclass1patientsonthesimilarityindex PreemptiveNon-preemptive 80%100.0%99.0%100.0%88.0%85%100.0%98.3%100.0%84.7%90%100.0%98.0%100.0%82.7% Table3-8. Impactofunitpatientdelaycostonthesimilarityindex PreemptiveNon-preemptive 200100.0%97.0%100.0%99.7%100.0%99.7%100.0%73.3%100.0%84.0%100.0%83.7%400100.0%97.0%100.0%99.7%100.0%99.7%100.0%74.3%100.0%84.7%100.0%84.3%800100.0%97.0%100.0%99.7%100.0%99.7%100.0%86.0%100.0%86.7%100.0%86.7% UnitofUP1tandUP2t:$/hourperpatient. Table3-9. Impactofmaximumallowableaveragetimeinsystemonthesimilarityindex PreemptiveNon-preemptive 6100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%7100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%8100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0% Unitof1and2:hourperpatient.

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Table 3-8 showsthattheunitpatientdelaycostdoesnotimpacttheaccuracyoftheHCTSCPmodelinthepreemptivecase.Butinthenon-preemptivecase,itsaccuracyindeterminingtheteamcongurationoftype2teamgoesdown,asunitclass1patientdelaycostorunitclass2patientdelaycostdecreases,andittendstooverestimatethetherequiredservicecapacityofthetype2teamby1level.Table 3-9 showsthatthemaximumallowableaveragetimeinsystemdoesnotmateriallyimpacttheaccuracyoftheHCTSCPmodelineitherthepreemptivecaseorthenon-preemptiveone.Insummary,theHCTSCPmodelwithDBAisecientandaccurateinsolvingthecapacityplanningproblems,particularlyinthepreemptivecase,e.g.,theEDapplication.Forthenon-preemptivecase,e.g.,theOCapplication,itsaccuracyintype2careteamrequirementsisnotasprecise. Inourwork,werstconsideredthecasewheretheclass1patientscanpreemptclass2patientsservedintype1E/Trooms,whichistypicalinemergencyroomsinhospitals. 66

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Wenotethatinourqueueinganalysis,weassumedexponentiallydistributedinterarrivalandservicetimestopreserveanalyticaltractability.Extendingourworktoconsidergeneralarrivalandserviceprocessesisapotentialareaforfutureresearch.Moreover,inourstudyweanalyzedsystemswherepatientsarecategorizedintotwoclasses.Althoughthisclassicationiswidelyusedinthehealthcareindustry,someclinicsandhospitalsfurtherclassifyeachoftheclassesintotwoormoresubclasses[ 100 ].Thepresenceofmultiplesubclassesineachpatientclassmagniestheproblemsizerapidlyandcannotbesolvedthroughgeneralnumericalmethods(e.g.,theGEortheAGEmethod).Therefore,developingapproximationproceduresforsuchsettingswouldbeoftheoreticalandpracticalinterest.Inourwork,wefocusedonthelong-termcapacityplanningofthehealthcareteams,treatingtheE/Troomsastheserversandassumingthattheserviceratesoftheroomsfordierentpatientclassesaregiven,androomallocationisxedovertheplanninghorizon.Formoredetailedanalysis,thesetoftasksrequiredforthediagnosis,monitoring,andtreatmentofapatientcanbemodeledusingaqueueing 67

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68

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Ineectiveallocationofexistingbedcapacityamongdierentmedicalserviceunitscanleadtoservicequalityproblemsforthepatientsalongwithoperationaland/ornancialinecienciesforthehospitals.AnarrivingpatientmaybedeclinedorplacedonholdbyanMCU,ifthereisnobedavailabletoaccommodatethepatientintheunit.Inthiscase,thepatientmaybesubjecttosamehealthrisksduetothenecessitytondanalternativehealthcareproviderorwaituntilabedbecomesavailable.Fromthehospital'sperspective,thepotentialrevenueiseitherlostordeferred,whichmayleadtosomenancialineciencies.Similarly,anarrivingpatientmaybeacceptedfortreatmentbutcanbeaccommodatedinanotherMCU(e.g.,anarrivingobstetricspatientcanbeboardedinneurosurgery).Inthiscase,thepatientmaybesubjecttosameunnecessaryhealthrisksduetothenecessitytobeboardedtogetherwithpatientswithmoredierent,possiblymoreserious,healthconditions.Fromthehospital'sperspective,thepotentialrevenueiscollectedbuttheoperationalcostsmayincrease,asthepatientmaybeboardedinanMCUwheretheserviceresourcesaremoreexpensive(e.g.,neurosurgerynursesmayhaveadditionalqualicationsthanobstetricsnursesandthemedicalequipmentinaneurosurgerydepartmentaretypicallymoreexpensivethantheonesinobstetrics). Toeectivelyutilizeexistingbedcapacity,hospitaladministrationcouldchoosefromamonganumberofalternativeplanningstrategiestotondanallocationofexisting 69

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Inthiswork,wefocusontherststrategyanddevelopamathematicalprogrammingformulationtoaddressthisproblem.Wealsodevelopeectivesolutionapproachestoobtainhighqualitysolutionsparticularlyforlarge-sized,realistictestinstances. Theremainderofthischapterisorganizedasfollows.InSection 4.2 ,wereviewtherelatedliterature.Section 4.3 presentsmathematicalprogrammingformulationforHBA.WedevelopthreeheuristicsolutionapproachesinSection 4.4 .InSection 4.5 ,wepresentresultsfromourcomputationalstudythatevaluatesthecomputationalperformanceoftheapproachesdevelopedinSection 4.4 .Section 4.6 includesadiscussionoftheresultsandsuggestsfutureresearchdirections. 46 ].Totakethestochasticnatureofhealthcaresystemsintoaccount,researchersutilizequeuingandsimulationmodelsindeterminetheappropriatebedcapacityconguration. Theapplicationofqueueingtheoryallowsfortheevaluationoftheexpected(long-run)performancemeasureofasystembysolvingtheassociatedsetofowbalanceequations.MackayandLee[ 80 ]evaluatethechoiceofmodelsforforecastingbedcapacityandsuggesttousecompartmentalowmodel,whichmodelspatientowthrougha 70

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Gorunescuetal.[ 43 ]modeladepartmentofgeriatricmedicineasanM=M=c=Kqueueingsystemtoinvestigatetheinterrelationshipsbetweenadmissionrates,lengthofstay,numberofallocatedbeds,andprobabilitythatanarrivingpatientisdeniedadmission.KaoandTung[ 63 ]presentanapproachforallocatingbedstocareunitsinahospitaltominimizetheexpectedpatientoverow,i.e.,therateatwhichpatientsaredeniedadmissionduetoinavailabilityofbedcapacity.TheymodeleachserviceasanM=G=1queueingsystemandusenormalapproximationincomputingnumberofpatientoverows.Thebedallocationproblemissolvedintwostages.Therststagedistributesthemajorityofbedssuchthatnogrossimbalancesinbedutilizationamongallcareunitsareobservedandapre-speciedfractionofpatientscanstayintheunitsdesignatedfortheunit.Thesecondstageusesmarginalanalysistoallocatetheremainingbedstominimizetheexpectedtotalpatientoverow. Discrete-eventsimulationisusefulfortheanalysisofsystemswithcomplexbehavior,includinghealthcaresystems.Discrete-eventsimulationhasbeenwidelyappliedinhealthcareservices[ 58 ]tostudytheinterrelationshipsbetweenadmissionrates,hospitaloccupancy,andseveraldierentpoliciesforallocatingbedstoMCUs.Harrisonetal.[ 52 ]constructasimulationmodelwherepatients'stayinhospitalsareclassiedintothreestages,whichrepresentdierentphasesofcareprovided.Theoutputobtainedfromthemodelmatchesthemeanandthevariabilityassociatedwithactualbedoccupancydata.Themodelisusedtoidentifydailyoccupancydistributions,studytrade-osbetweenoverowandbedcapacitylevels,andinvestigatetheeectsofvariouschanges.AkkermanandKnip[ 6 ]useMarkovchainapproachtospecifythenumberofbedsneeded 71

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84 ]usediscrete-eventsimulationtoinvestigatetheinterrelationshipsbetweenbedoccupancy,averagenumberofpatientdeferred,anddierentbedallocationandoperatingpolicies.Harper[ 51 ]developsasimulationmodelfortheplanningandmanagementofhospitalbeds,operatingtheaters,andworkforceneeds.Themodelcapturesthecomplexityofhealthcaresystemsbyincorporatingthevariabilityforeachpatientgroupsuchasmonthly,daily,andhourlydemandaswellasthedistributionsoflengthofstayandoperationtimes.Kimetal.[ 68 ]analyzeanintensivecareunitwith14bedsanddevelopasimulationmodeltoevaluatedierentbed-reservationschemestoreducethenumberofcancelledsurgeries. Somepapersconsiderthehierarchicalrelationbetweencareunits.Forexample,afteramother-to-bedeliversherchildinthelaboranddeliveryunit,sheshouldbemovedtothepostpartumunitforrecovery.Whenthecapacitydownstreamisinsucient,patientsareforcedtostayatthecurrentcareunitswithtypicallymoreexpensiveequipmentblockingthecapacityattheseupstreamcareunits.Totaketheinteractionsamongcareunitsinahospitalintoaccount,CochranandBharti[ 30 ]rstapplyqueueingnetworkmethodology(withoutblocking)tondabalancedbedallocation,whichisobtainedthroughtrial-and-errorwork.Then,theyusesimulationanalysistoestimatetheblockingbehaviorandpatientsojourntimes.Galv~aoetal.[ 39 40 ]applyathree-levelhierarchical,capacitatedmodeltodeterminethecapacityrequiredinperinatalhealthcarefacilities,whichiscategorizedintothreelevels:basicunits,maternityhomes,andneonatalclinicswhereintensivecareunitforbabiesisavailable. Inourwork,weutilizeadierentapproachbyintegratingresultsfromqueueingtheoryintoanoptimizationframework.Specically,wemodeleachMCUinahospitalasanM=M=c=cqueueingsystemtoestimatetheprobabilityofrejectionwhentherearecbedsintheunit.Wethendevelopanoptimizationmodeltoallocatetheaggregatebed 72

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minDXi=1jij 73

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4{1 )minimizesthetotaldeviationofthebedoccupancyforeachoftheMCUsfromtheoverallaveragebedoccupancyforthehospital.Constraint( 4{2 )limitsthetotalnumberofbedsallocatedamongtheMCUstothenumberoftotalbedsavailableinthehospital.Constraintset( 4{3 )representstheprobabilityofrejectingpatientsarrivingtoserviceiasafunctionofarrivalratei,servicerateiandbedcapacityxiforservicei.Constraintset( 4{4 )representsthebedoccupancyoftheserviceiasafunctionofeectivearrivalrate(1pi)i,servicerateiandbedcapacityxiforeachservicei.Constraint( 4{5 )speciestheoverallaveragebedoccupancyforthehospital.Constraintsets( 4{6 )and( 4{7 )imposethelowerandupperboundsonthebedcapacityandthebedoccupancyforeachservicei,respectively.Finally,constraint( 4{8 )ensuresthatthedecisionvariablesarenon-negativeintegers. HBAisadicultnonlinearbinaryintegerprogrammingproblemwithnonlinearconstraints.Notethatconstraintset( 4{3 )involvesthefactorialfunctiononxiandthesummationoftheterm(i=i)m Asaresult,wecanobtainanequivalentlinearbinaryintegerprogrammingproblemthatcanbestatedasfollows: minDXi=1uiliXj=1ijyij 74

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Wehavefoursetsofadditionalconstraints.Constraintset( 4{15 )ensuresthatonlyonesplittingvariabletakesthevalueofone.Constraintsets( 4{16 )and( 4{17 )computethedeviationofthebedoccupancyforeachserviceifromtheoverallaveragebedoccupancyforthehospital.Constraintset( 4{18 )ensuresthatthedecisionvariablestakebinaryvalues.Finally,constraintset( 4{19 )providesinformationofthediscreteoptionsofxij. 41 53 ].Inparticular,twomembersofthecurrentpopulationofsolutionsarechosenrandomlyandusedtogeneratenewospring 75

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4-1 depictsthepseudo-codeoftheGA.Apreprocessingstepensuresthatatestinstanceisfeasiblebyverifyingthevalidityofthefollowinginequalities: where(li)representsthebedoccupancywithbedcapacityliofservicei. endGeneticAlgorithm. Figure4-1. Pseudo-codeofthegeneticalgorithm GAstartsbygeneratingasetofdistinctfeasiblesolutionstoformaninitialpopulationofsolutions.Thesesolutionsaregeneratedthroughanoccupancy-drivenapproach,andthenadjustedtofeasiblebyarandom-rectiedapproach(seeFigure 4-2 ). Theoccupancy-drivenapproachrsttreatseachMCUasanM=M=cqueueingsystem,andthenallocatesanumberofbedstotheMCUsuchthatthebedoccupancyiscloseto(seeFigure 4-3 ).Asitcanbeobservedfromconstraintset( 4{3 ),specifyingtheprobabilityofrejectingapatientfromanMCUthatismodeledasanM=M=c=cqueueingsystemrequirestheknowledgeofthenumberofbedsallocatedtothatparticularMCU.SinceourobjectiveistogeneratesomesolutionsfortheinitialpopulationofGA,rather 76

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Figure4-2. Pseudo-codeofthepopulationgeneratingprocedure thanspendingtimetondthecorrespondingbedcapacityforanM=M=c=cqueueingsystem,wemodeltheMCUasanM=M=cqueueingsystemtoinitializethebedcapacity. Figure4-3. Pseudo-codeofoccupancy-drivenallocation Thesolutionsgeneratedfromtheoccupancy-drivenapproachmaynotsatisfyconstraintsets( 4{2 ),( 4{6 )and( 4{7 ),i.e.,constraintsetsassociatedwiththetotalnumberofbedsavailable,lowerandupperboundsandbedoccupancy(ofanM=M=c=cqueueingsystem).Weusearandom-rectiedproceduretone-tunetheinitialbedallocation.Figure 4-4 showsthepseudo-codeoftherandom-rectiedprocedure.WerstrevisethebedcapacityofeachMCUsuchthattheconstraints( 4{6 )and( 4{7 )aresatised.Then,whenthetotalnumberofbedsallocatedisgreater(less)thanB,anMCUischosenrandomlyanditsbedcapacityisdecreased(increased)byone,ifthisoperationdoesnotviolateconstraint( 4{6 ).ThestepofchoosinganMCUrandomlyandadjustingitscapacityaccordinglyisrepeateduntilthetotalnumberofbedsallocatedisequaltotheoverallbedcapacityavailable,B. 77

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Figure4-4. Pseudo-codeofrandom-rectiedprocedure Aftertheinitialpopulationisformed,GAstartstoproduceospring.Twotypesofgeneticoperatorsareconsideredinthiswork;oneissimplesingle-pointcrossover,andtheotherismutation.Insimplesingle-pointcrossover(seeFigure 4-5 ),werstrandomlyselecttwoparentsfromthepopulation,andthenswapbedcapacitybetweentherandomlyselecteddD=2eMCUs.Thatis,iftheMCUiisselected,thebedcapacityofMCUioftherstparent'schromosomeisswappedwiththebedcapacityofMCUiofthesecondparent'schromosome.Theresultingsolutionsaretheospringsolutionsproducedbytheselectedparentsolutions.Figure 4-6 showsanexamplewithfourMCUsand100beds,wherethesecondandthethirdMCUsarechosentoswaptheirbedcapacity.Notethatthecrossovermayproducedanospringwhichdoesnothaveachromosomespecifyingafeasiblesolution.Inthiscase,therandom-rectiedapproachisemployedtoadjustthebedcapacityallocation. Thesecondgeneticoperator,mutation,isinvokedifthecurrentbestsolutionisnotimprovedthroughouttheevolutionofapre-speciednumberofconsecutivegenerations.Themutationemploystheoccupancy-drivenapproachwiththeaverageoccupancy 78

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endCrossover. Figure4-5. Pseudo-codeofcrossoverprocedure Figure4-6. Exampleofcrossover ofthecurrentbestsolutiontoproduceanospring.Again,thismutatedospringisadjustedthroughtherandom-rectiedproceduretoensurefeasibility.Then,thevalueofNumGenNotImproveissettozero. Iftheospringproducedfromeithercrossoverormutationproceduredoesnotexistinpopulationandisbetterthanthecurrentworstsolutioninpopulation,thenthecurrent 79

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Figure4-7. Pseudo-codeofthemutationprocedure worstsolutioninthepopulationisreplacedwiththeospringtoformanewgenerationwiththeotherexistingsolutions.Otherwise,i.e.,eithertheospringexistsinpopulationalreadyortheospringisnotbetterthanthecurrentworstsolutioninthepopulation,thisospringisignoredandthenextospringisproducedusingtheproceduredescribedabove.Notethatthecurrentbestsolutionisalsoupdatedifthenewospringhasabetterobjectivefunctionvaluethanthatofthecurrentbest. 36 ]thatiswidelyusedforcombinatorialoptimizationproblems.EachGRASPiterationconsistsoftwophases:constructionandlocalsearch.Figure 4-8 depictsthepseudo-codeforGRASP.Theconstructionphasecreatesafeasiblesolution,whoseneighborhoodisexploredbythelocalsearchphasetondalocallyoptimalsolution.Thealgorithmstopsafterapre-speciednumberofiterationsisexecuted,whichisdenotedbyMaxNumIterationsinFigure 4-8 endGRASP. Figure4-8. Pseudo-codeofGRASP 80

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4-9 ,wheretheejinline 22 representsazerovectorexceptthejthelementequalsone.Lines 15 through 20 inFigure 4-9 buildarestrictedcandidatelist(RCL),whichrecordsthesetofMCUsforwhichaddingonemorebedtotheMCUdoesnotviolatethefeasibilityandhasthepotentialtoimprovetheobjectivefunctionvalueatanacceptablelevel.Thethresholdofincrementontheobjectivefunctionvalueiscontrolledbytheparameter2f0;1gonline 17 inFigure 4-9 .AnMCUisincludedinRCL,ifaddingonemorebedtotheunitisfeasible,andtheincrementontheobjectivefunctionvalueisnotgreaterthanmin+(maxmin),whereminandmaxrepresenttheminimumandmaximumincrementsonobjectivefunctionvalueafterincorporatingonemorebedtocurrentsolution,respectively.Notethatthelowertheis,thegreediertheprocedureis.AbedisaddedtoanMCUselectedrandomlyfromRCL,untilthetotalnumberofbedsallocatedequalsB. Duringthelocalsearchphase,theneighborhoodofthefeasiblesolutioncreatedintheconstructionphaseisfullyinvestigatedtondthelocaloptimum.AneighborsolutionisproducedbyswappingonebedfromoneMCUtoanother,ifthisswapdoesnotdestroythefeasibilityofthesolution.Thepseudo-codeofthelocalsearchprocedureisillustratedinFigure 4-10 .Here,weadoptthebest-improvingstrategythatisweevaluateallfeasibleneighborsandchoosetheneighborthatimprovestheobjectivefunctionvaluethemost. 4-11 .OneachiterationoftheHA,asetofelitesolutionsisgenerated,incontrasttoasingleinitialsolutionusedbyGRASP.Then,anospringis 81

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Figure4-9. Pseudo-codeofgreedyrandomizedconstructionprocedure producedusingthecrossoverprocedureofGAandimprovedbythelocalsearchprocedureofGRASP.Theprevioustwostepsarerepeated,untilapre-speciednumberofiterationsarecompleted.ThecharacteristicoftheHAisthatoneachiterationthethresholdforthegreedyrandomizedconstructionphaseisupdatedbyadecreasingfunctionp(n),i.e.,decreasesasthenumberofiterations,n,increases.Inthisway,theHAcantestmorethanonesettingofandreducetheprobabilityofconvergingtoalocaloptimumsolutionprematurely. Toformanelitesetofsolutions,werstusetheGreedyRandomizedConstructionprocedureofGRASPtogenerateNumGRASPdistinctsolutions,andthenchoosethebestESizeofthem,whereNumGRASP>ESize.Thepseudo-codeispresentedinFigure 4-12 82

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Figure4-10. Pseudo-codeoflocalsearchprocedure endHybrid. Figure4-11. Pseudo-codeofHA 83

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Figure4-12. Pseudo-codeofelitesetgenerationprocedure wherefAdenotestheobjectivefunctionvalueobtainedbyusingapproachA,whereA2fGA;GRASP;HAg. Inourstudy,weconsiderthreesettingseachwithdierentinstancesizestoevaluatetheimpactofthesizeoftheinstanceontheperformanceoftheproposedsolutionapproaches.Theproblemsizeisvariedwiththenumberoftotalbedsavailableinthehospital,B,andnumberofMCUsinthehospital,D.ForthetotalnumberofbedsandthenumberMCUsavailableinthehospital,weconsiderthreelevelsthatcorrespondtosmall,medium,andlargesizedhospitals.Inparticular,weconsiderhospitalswith750,1,000or1,250bedsand30,40or50MCUs.Foreachsetting,wegenerate30randominstances.EachrandominstanceisobtainedbygeneratingparametersthatcorrespondtopatientarrivalandserviceratesalongwithlowerandupperboundsonthenumberofbedsavailableineachMCU.Specically,eachparameterisobtainedbyusingtheformula:(meanvalue)u,whereuisdrawnfromthedistributionU[0:4;1:6].Themeanvaluesoftherandomparametersarelistedasfollows: 1.Meanarrivalrate(i):10persons/perunittimeforeachMCUi; 2.Meanservicerate(i):0.5persons/perunittimeforeachMCUi; 3.Meanlowerbound(li):10bedsforeachMCUi;and 84

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Last,theminimumbedoccupancy,,is70%forallMCUallproblems.TheexperimentisimplementedonaworkstationwithtwoPentium43.2GHzprocessorand6GBofmemory. Toobtainanear-optimalsolutionasacomparisonbasis,weuseCPLEXtoobtaintheoptimalsolutionsforthetestinstances.Foreachtestinstance,CPLEXisstoppediftherelativestoppingtoleranceof0.01%issatised,orCPUtimeof3,600secondsisused.Table 4-1 reportstheresultsobtainedbyCPLEX.Asweexpect,thenumberofinstancesforwhichCPLEXcandetermineandverifytheoptimalsolutionwithinonehourofCPUtimedecreasesastheproblemsizeincreases.Moreover,fortheinstanceswithsmallproblemsize,CPLEXtakesmorethan1,800secondsonaveragetosolveoneinstance. 4-2 reportstheresultsofGA,wheretheinformationisclassiedintotwolayers.Therstlayerdistinguishestheinstanceswithdierentsizes,andthesecondspeciestheparametersettingofGA,includingnumberofgenerationsandpopulationsize.Notethatiftherelativeerrorofaninstanceislessthanzero,itisreplacedbyzerowhencomputingtheaveragerelativeerrorofthecorrespondinginstance. Ingeneral,GAperformsquitewellintermsofaveragerelativeerrorandCPUtime.Foranyofthe90instances,GAspendslessthan2secondstondasolutionandtheaveragerelativeerrorislessthan2%.Furthermore,formorethanonethirdofthetestinstances,GAndssolutionsbetterthanCPLEX.Forexample,forthefourthlarge-sizedinstance,GAndsasolutionwithanobjectivefunctionvaluewhichisabout5.6%lowerthanthatofthesolutionfoundbyCPLEX. 85

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Near-optimalsolutionsobtainedbyusingCPLEX InstanceObjectiveTime(sec.)ObjectiveTime(sec.)ObjectiveTime(sec.) 10.1964370.26913180.5298>3,60020.613240.4122>3,6001.0745>3,60030.2574>3,6000.6417>3,6000.5083>3,60040.28691,9720.3864>3,6000.7238>3,60050.4828>3,6000.4141>3,6001.0353>3,60060.4588210.7636>3,6000.2228>3,60070.28171,3690.3180>3,6000.5841>3,60080.24847090.5691>3,6000.4073>3,60090.13512,1110.4530>3,6000.6679>3,600100.45581100.3737>3,6001.1034>3,600110.34471,2770.3347>3,6000.5721>3,600120.4320470.4972>3,6000.6960>3,600130.2872>3,6000.6381>3,6000.4622>3,600140.458940.7216>3,6000.7431>3,600150.5883>3,6000.6496>3,6000.4568>3,600160.28483750.8077>3,6000.9320>3,600170.2751>3,6000.5183>3,6000.5609>3,600180.1942>3,6000.20225430.5702>3,600190.3408>3,6000.6919290.7131>3,600200.260080.4640>3,6000.6928>3,600210.3262>3,6000.6155>3,6001.0242>3,600220.4457>3,6000.52051,7270.6448>3,600230.181380.4363>3,6000.4886>3,600240.3991>3,6000.1789>3,6000.7533>3,600250.6148>3,6000.3602>3,6000.7442>3,600260.3832>3,6000.7231>3,6000.4640>3,600270.8989120.7193>3,6000.5765>3,600280.33927300.5233>3,6000.7700>3,600290.13771,7700.6959>3,6000.4138>3,600300.4448130.48335771.0881>3,600 No.ofinstancessolvedbyCPLEX1850 86

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SolutionsobtainedbyGA MediumLargesize No.of6000 60006000 800080008000 100001000010000generationsPopulation30 6090 4080120 50100150size Err.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeInstance (%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.) 1 0.000.560.000.570.000.580.001.000.001.000.001.026.261.566.261.606.261.682 9.280.589.280.609.280.623.861.013.861.043.861.06-2.101.60-2.101.59-2.101.673 0.000.570.000.580.000.600.001.030.001.040.001.060.001.560.001.590.001.674 0.000.560.000.570.000.571.741.001.741.031.741.04-5.581.52-5.581.56-5.581.645 0.000.600.000.610.000.610.000.980.000.990.001.010.691.560.691.580.691.656 0.000.550.000.560.000.573.091.043.291.073.311.090.001.580.001.620.001.707 0.000.570.000.580.000.590.000.990.001.010.001.020.001.590.001.640.061.738 0.000.550.000.560.000.570.001.031.411.060.001.090.001.580.001.620.001.709 0.000.570.000.570.000.580.001.030.001.030.001.040.881.601.281.691.871.7710 0.000.560.000.580.000.580.440.990.441.020.441.047.061.607.131.668.201.7411 0.000.570.000.580.000.590.001.020.001.040.001.060.001.560.001.610.001.6812 0.000.570.000.590.000.620.001.000.061.020.061.035.781.585.781.605.781.6613 0.000.560.000.560.000.570.900.990.901.000.901.020.001.570.001.600.001.6814 3.450.553.450.563.450.579.450.999.451.009.451.02-0.111.58-0.111.62-0.111.7115 0.000.570.000.580.000.580.001.010.001.020.001.040.001.540.001.550.001.6316 0.000.560.000.570.000.590.001.000.001.030.001.041.821.562.001.592.351.6717 0.000.570.000.570.000.580.001.010.001.030.001.051.091.601.091.641.091.7318 0.000.580.000.610.000.620.001.040.001.050.001.080.591.600.591.660.411.7619 0.000.570.000.570.000.583.431.042.491.084.681.110.661.580.661.610.671.7020 0.000.600.000.620.000.640.001.019.341.029.341.050.911.581.041.614.341.6921 0.000.570.000.570.000.590.001.020.001.050.001.070.271.550.271.590.001.6822 2.060.560.000.580.000.590.000.970.000.980.001.000.321.600.321.640.321.7323 0.010.560.010.560.010.560.001.030.001.060.001.091.011.611.081.661.101.7724 4.830.554.830.564.830.580.001.010.001.010.001.033.401.523.401.543.401.6225 2.820.573.480.583.550.590.001.010.001.040.001.060.001.530.001.551.101.6226 0.000.570.000.580.000.597.201.037.201.057.211.08-2.241.55-2.241.58-2.241.6527 0.000.560.000.570.000.58-2.061.00-2.061.02-2.061.05-0.621.561.001.601.961.6928 0.000.580.000.590.000.600.001.020.001.050.001.08-3.761.56-3.691.58-2.311.6729 0.000.580.000.600.000.625.401.005.401.025.401.040.001.530.001.570.001.6330 4.950.595.030.605.220.620.001.020.001.040.001.072.451.603.041.642.501.73 Min. 0.000.550.000.560.000.56-2.060.97-2.060.98-2.061.00-5.581.52-5.581.54-5.581.62Avg. 0.910.570.870.580.880.591.181.011.521.031.551.051.111.571.191.611.401.69Max. 9.280.609.280.629.280.649.451.049.451.089.451.117.061.617.131.698.201.77 No.of13 1313 111011 121010instancesGAndsbettersoln.thanCPLEX

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4-3 displaystheresultsofGRASP.AsisobvioustoseethatGRASPtakesCPUtimeslongerthanGAtoobtainsolutions.GRASPspends3.1secondsinaveragetoobtainasolutionoftheproblemwithsmallsize,incontrasttoGA's1.6secondswithrespecttotheproblemwithlargesize.Ingeneral,theaccuracyofGRASPandGAdoesnotappearsignicantlydierent,ifwecomparethebestresultsfromeachofthem.Forexample,intheproblemwithmediumsize,therelativeerrorsofGRASP(with=0.25)are1.37%and8.24%inaverageandmaximum,respectively,andGA(withpopulationsize=40)are1.18%and9.45%,respectively.Furthermore,among30instanceGRASP(with=0.25)ndsresultsbetterthanCPLEXfor9instances,comparedto11instancesofGA(withpopulationsize=40). Table 4-3 presentsthetendencythattheloweris,thelowertheaveragerelativeerroris.However,alowvalueofhasthedrawbackthatitmayleadthesolutiontoalocaloptimalsolutionandresultinalargeerror.Forexample,theinstance20inproblemwithmediumsizehasrelativeerror8.24%whenis0.25.Incontrast,theinstancehaslessrelativeerror,3.77%,whenisincreasedto0.35. InGAandGRASP,wedoobservethatsomeinstanceshaverelativeerrorgreaterthan8%,suchastheinstance14oftheproblemwithmediumsizeinTable 4-2 ,andtheinstance20oftheproblemwithmediumsizeinTable 4-3 .Toreducetheerrorsofthoseinstances,wecombinesthesetwoalgorithmstodevelopahybridization. 88

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SolutionsobtainedbyGRASP Medium Largesize No.of600 600600 800800800100010001000starts0.25 0.300.35 0.250.300.35 0.250.300.35 Err.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeInstance (%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.) 1 0.003.000.002.960.002.940.008.900.008.840.009.396.3322.786.7622.518.5624.192 0.983.242.813.182.813.154.369.104.578.957.919.21-2.1023.72-2.1023.54-2.1024.003 0.002.920.002.900.002.880.009.830.009.750.0910.390.0023.310.0622.981.7324.554 0.003.220.003.210.003.202.769.535.279.525.2710.03-5.5823.50-5.5823.34-5.5824.845 0.003.200.003.141.083.130.009.210.009.130.009.360.4724.360.9524.211.2725.026 -0.783.27-0.783.23-0.783.222.889.653.319.5411.1910.020.0022.300.0022.183.0623.717 0.003.090.003.050.003.050.009.840.009.760.0010.340.4922.440.9422.102.5223.938 0.003.130.003.080.003.060.009.350.009.200.149.420.0024.020.1723.851.4825.319 0.003.540.003.520.003.511.149.631.489.532.3210.192.6822.473.0221.985.0422.5910 0.002.930.002.930.002.930.449.720.449.640.4410.116.0822.778.6522.4419.5923.9511 0.003.030.002.990.002.970.009.920.009.850.0010.360.0024.210.0023.900.0025.4312 0.003.090.093.061.113.030.069.580.069.390.069.785.7823.755.7823.525.8723.9313 0.003.170.003.130.003.130.909.710.909.660.9010.150.0023.870.0023.660.0024.8014 -0.143.39-0.143.35-0.143.327.529.607.889.589.0010.250.5023.254.8123.0317.5824.2115 0.003.120.003.100.003.110.189.970.189.800.5710.120.0023.550.0023.440.0024.8716 0.003.250.003.212.083.200.429.450.099.322.389.633.2023.913.9023.663.8625.1417 0.003.450.003.410.003.360.009.840.009.750.7110.030.5123.972.1423.854.8325.0618 0.003.410.003.373.883.330.009.170.009.010.009.550.0923.441.3023.194.8824.7119 0.002.990.002.950.002.941.738.962.668.864.929.170.6621.970.7521.781.4923.2120 0.003.220.003.160.003.138.249.366.019.193.779.504.4323.144.4622.868.0823.2021 0.003.330.003.330.003.290.009.520.449.3718.199.91-1.4823.21-1.4222.927.3724.8322 0.503.100.403.060.003.04-1.279.54-1.279.46-1.149.741.0821.520.9721.085.5422.3323 0.013.170.013.170.013.140.039.390.399.272.239.521.2823.232.5322.9511.6024.4224 4.833.194.833.144.833.130.0010.010.009.870.0010.513.2724.743.2824.543.3026.1625 5.513.206.173.195.833.170.009.680.009.530.009.816.2622.882.7722.536.5424.1326 0.003.220.003.180.003.174.569.494.599.409.399.67-0.5023.29-0.5023.16-0.5024.5727 -7.043.31-7.043.26-7.043.25-2.069.53-2.069.42-2.069.882.6323.571.2423.281.3724.5328 0.003.134.003.0611.953.030.059.110.669.002.159.298.3823.498.0623.088.8424.0529 0.003.010.002.980.002.935.689.775.529.706.2310.290.0024.000.0023.920.0025.3630 3.742.951.112.933.982.910.009.500.259.346.919.881.9822.792.0122.255.6924.07 Min. -7.042.92-7.042.90-7.042.88-2.068.90-2.068.84-2.069.17-5.5821.52-5.5821.08-5.5822.33Avg. 0.523.180.653.141.253.121.379.531.499.423.169.851.8723.322.1523.064.6724.37Max. 5.513.546.173.5211.953.518.2410.017.889.8718.1910.518.3824.748.6524.5419.5926.16 No.of15 1312 986 875instancesGRASPndsbettersoln.thanCPLEX

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Constant:p1(n)=0:3;Linear:p2(n)=1:00:7n N;andNonlinear:p3(n)=1:00:7r N Toprovideabenchmark,weincludetheconstantfunctionp1(n)inourexperiment,whereissetto0.3oneachiterationnforn=1,2,...,bD=10c.Usingfunctionp2(n),welinearlydecreaseto0.3asthenumberofiterationsincrease.Usingfunctionp3(n),wedecreaseto0.3inanon-linearfashionovertheiterations.Notethatvaluesgeneratedbyfunctionp3(n)isnolargerthanthosegeneratedbyfunctionp2(n)foreachnasshowninFigure 4-13 Figure4-13. Theupdatingfunctionsof 4-4 .Ineachtestinstance,therstcolumnreportstherelativeerrorsandCPUtimesobtainedbyusingtheupdatingfunctionp1(n),andsoon.Forthemedium-sizedtestinstances,theaverage(maximum)relativeerrordecreasesfrom1.28%(9.34%)to0.35%(4.56%),whenp2(n)isusedinsteadofp1(n).Thefunctionp3(n),whichhassmallerthanp2(n)ateachiterationn,doesnotappeartomakeHAperformbetterthantheHAwithp2(n).Forexample,theaveragerelativeerrorforlarge-sizedinstancesincreasesfrom0.35%to0.44%,ifthep2(n)isreplacedbyp3(n).Thisisdierent 90

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SolutionsobtainedbyHA Medium Largesize No.of3 33 444555iterationsNo.of6000 6000 600080008000 80001000010000 10000generationsp1(n) 3030 404040 505050 Err.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeInstance (%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.) 1 0.007.940.007.930.007.920.0027.650.0027.660.0027.356.2671.110.0071.860.0072.512 2.817.650.007.710.007.663.8625.310.0025.800.0026.18-2.1062.19-2.1064.49-2.1063.263 0.007.180.007.700.007.440.0024.220.0024.560.0024.790.0077.700.0075.530.0077.464 0.008.070.008.170.007.951.7425.751.7425.521.7425.84-5.5874.41-5.5873.71-5.5874.295 0.007.290.007.500.007.350.0025.760.0023.710.0025.260.6966.540.6965.090.6965.696 -0.588.000.008.140.008.052.5525.060.0025.680.0025.810.0077.340.0077.190.0077.767 0.007.390.007.740.007.330.0027.990.0028.060.0028.290.0070.420.0073.270.0072.768 0.007.940.007.070.007.910.0024.650.0024.630.0024.740.0072.010.0073.610.0074.519 0.008.180.008.360.008.130.0026.050.0026.450.0026.412.6870.812.6869.480.0070.4510 0.007.040.007.270.007.080.4425.870.4425.310.4425.702.6067.49-1.3567.591.5267.8311 0.007.800.007.930.007.750.0026.490.0026.420.0026.680.0075.010.0074.540.0075.3012 0.007.710.007.900.007.820.0626.630.0026.520.0626.805.7876.06-1.5775.61-1.5777.0113 0.008.220.008.260.008.220.9026.910.9026.800.9026.810.0072.490.0071.260.0072.3014 -0.147.460.167.471.927.358.6525.372.8325.663.7625.74-0.1169.44-0.1170.55-0.1170.0615 0.007.330.007.710.007.400.0027.190.0027.200.0027.530.0078.020.0077.320.0078.3016 0.007.600.007.770.007.630.0023.770.0023.37-0.6824.461.6267.230.0067.170.0068.1217 0.008.140.008.400.008.150.0025.360.0025.250.0025.530.0069.200.7870.690.7871.0118 0.007.880.008.300.008.070.0027.850.0027.920.0028.140.0072.930.0073.510.0073.4119 0.007.460.007.880.007.511.4625.200.0025.070.0025.090.6670.790.0070.190.6671.4620 0.007.450.007.760.007.549.3426.680.0027.300.0027.424.3472.950.0074.364.3474.2821 0.007.400.007.680.007.470.0025.820.0025.030.0025.740.0066.710.2769.060.1468.9522 0.007.530.007.820.007.56-1.0526.080.0025.70-0.1426.350.0067.370.0068.340.0067.7223 0.016.190.016.380.016.170.0025.550.0026.180.0026.351.0169.361.0173.450.0072.9424 4.837.574.837.954.837.550.0028.010.0028.180.0028.523.4072.413.4071.363.4073.0825 2.827.802.828.072.827.830.0026.610.0026.600.0026.840.0071.320.0070.610.0071.8026 0.007.790.008.220.007.904.0325.864.5625.494.2325.74-2.2478.86-2.2477.07-2.2478.6227 -6.057.37-5.917.32-6.707.08-2.0625.52-2.0625.51-2.0625.78-1.0675.35-1.0675.18-1.0675.1328 0.007.470.007.810.007.620.0023.740.0024.200.0024.46-3.7670.79-3.7670.87-3.7671.5329 0.007.920.008.230.007.975.4026.840.0027.172.0627.260.0078.760.0078.240.0079.1230 3.897.330.007.690.007.620.0024.800.0025.400.0025.350.0069.440.0068.750.0069.22 Min. -6.056.19-5.916.38-6.706.17-2.0623.74-2.0623.37-2.0624.46-5.5862.19-5.5864.49-5.5863.26Avg. 0.487.600.267.800.327.631.2825.950.3525.950.4426.230.9771.820.2972.000.3872.53Max. 4.838.224.838.404.838.229.3428.014.5628.184.2328.526.2678.863.4078.244.3479.12 No.of15 1515 111314 151716instancesHBndsbettersoln.thanCPLEX

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HAoutperformsbothGAandGRASPintermsofsolutionquality.Therelativeerrorislessthan0.5%inaverageand5%inmaximumforallproblemswhetherp2(n)orp3(n)isused.Furthermore,HA(withp2(n)orp3(n))ndsbettersolutionsthanCPLEXforabouthalfoftheinstances,comparedtoGAorGRASP'sonethird.Astothecomputationalrequirement,HAspendslessthan80secondstoobtainasolutionforthelarge-sizedinstances,whichislargerthaneitherGAorGRASP,butisstillatanacceptablelevelfromapracticalperspective.Fromthese,wecanconcludethatHAisaneectiveandecientsolutionapproachfortheHBAproblem. Inourwork,eachMCUismodeledasanM=M=c=cqueueingsystem.Futureresearchcanconsiderdierentqueueingsystemcongurationswithgeneralarrivalandservice 92

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93

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Inahealthcareservicefacility,whentheemergencyroom(ER)isfullorallintensivecarebedsareoccupied,hospitalssendoutdivertstatus.Whenahospitalisondivertstatus,incomingpatientsmightbesenttohospitalswhicharefartherawayorkeptatthehospitalswheretheycurrentlyarethatmaynotabletoprovideadequateservice.Toacriticalpatient,theconsequenceofdivertstatuscanbethedierencebetweenlifeanddeath. Thepurposeofthisworkistoconstructafacilitylocationmodel,whichsimultaneouslydeterminesthenumberoffacilitiesopenedandtheirrespectivelocationsaswellasthecapacitylevelsofthefacilitiessothattheprobabilitythatallserversinafacilityarebusydoesnotexceedapre-determinedlevel.Inotherwords,wewanttolocateERservicesonanetworkanddeterminetheirrespectivecapacitylevelssuchthattheprobabilityofdivertingpatientsisnotlargerthanaparticularthreshold. Theremainderofthischapterisorganizedasfollows.InSection 5.2 ,wereviewtherelatedliterature.Section 5.3 presentsmathematicalprogrammingformulationforERSFLCP.Section 5.4 detailstheLagrangianrelaxationalgorithmthatweproposefortheERSFLCPproblem.InSection 5.5 ,wepresentresultsfromourcomputationalstudythatevaluatesthecomputationalperformanceoftheLagrangianrelaxationalgorithm.Section 5.6 includesadiscussionoftheresultsandsuggestsfutureresearchdirections. 94

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Thereisarichbodyofliteraturethatdeveloprobustfacilitylocationmodelsunderuncertaintytohedgetherandomnessoncosts,demands,andtraveltimesutilizingtherobustorstochasticoptimizationapproach.Snyder[ 103 ]presentsadetailedreviewofonstochasticandrobustfacilitylocationmodels. Intheareaofhealthcareapplication,Beraldietal.[ 12 ]investigatetheproblemofcharacterizingtheoptimallocationsofemergencymedicalservicesitesandnumbersofemergencyvehiclesrequiredforeachsite.Theyconsidertheproblemformulationinastochasticoptimizationsetting,wheretheabilitytocovertherandomrequestsofemergencyserviceatdemandpointsisrestrictedbyasetofprobabilisticconstraints.Specically,theprobabilisticconstraintsensurethattheprobabilitythatthenumberofvehicleslocatedatafacilitycancovertherandomservicerequestisgreaterthanaprescribedprobabilityvalue.Somepapersconsiderthehierarchicalrelationbetweenfacilitiesinahealthcaresetting.Forexample,Koizumietal.[ 70 ]classifymentalhealthcaresystemintolevelsofextendedacutehospitals,residentialfacilitiesandsupportedhousing,andpatientsowthroughthelevelsaccordingtotheirhealthconditions.Becauseofthehierarchicalstructureofhealthcareservice,thecapacityrequirementsofunitsinahigherlevelofhierarchyusuallycorrelatewiththeunitsinalowerlevel.Galv~aoetal.[ 39 40 ]formulateahierarchicallocation-allocationmodeltodeterminethecapacityrequired 95

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Anotherbranchofliteraturethatisrelatedtoourworkdevelopsfacilitylocationmodelswhichrequirebackup(multiple)coverageforeachdemandpoint.SnyderandDaskin[ 104 ]consideraP-median-basedmodelthatminimizestotalcostsassociatedwithalocation-allocationplanandtheexpectedfailurecost.Aseachdemandpointisassignedprimaryandbackupfacilities,theexpectedfailurecostisquantiedbytheadditionaltransportationcostincurredtocoverthedemandbytheback-upfacility.Theproblemisformulatedasa0-1integerprogrammingformulationandsolvedbytheLagrangianrelaxationalgorithm.Jiaetal.[ 56 ]presentadetailedreviewoftraditionalfacilitylocationmodelsandproposeageneralfacilitylocationmodelsuitedforlarge-scaleemergencies.Intheirmodel,ademandpointisconsideredtobecoveredifapre-speciednumberoffacilitiesareassigned. Toaddresstheissueofservicequality,somepapersincorporatequeuingsystemsintofacilitylocationmodelstoconsidertherandomnessonavailabilityofserversandfocusonreducingthedemandlostduetotheshortageofcapacityorsystemcongestion.MarianovandReVelle[ 81 ]formulateamaximalavailabilitylocationmodel,whichusesaM=M=c=cqueueingsystemtomodeltheserveravailabilityofademandpoint.Themodelwantstolocateasetofambulancessuchthatthedemandcoveredismaximized,whereademandpointisconsideredtobecovered,iftheprobabilitythatthereisanambulancenearbyandavailableisgreaterthanathreshold.Theyshowhowtotransformthenonlinearqueueingexpressiontoanequivalentlinearone,andsolvetheproblembyusingacommercialsolver(LINDO).Bermanetal.[ 13 ]modeleachfacilityasanM=M=1=aqueueingsystem,whereaisthemaximumnumberofcustomersallowedinthefacility,andconsiderafacilitylocationproblemwiththeupperboundsontheamountofdemandlostduetoinsucient 96

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14 ]alsoinvestigatethefacilitylocationproblemundertheobjectiveofmaximizingcaptureddemand.They,again,modeleachfacilityasanM=M=1=aqueueingsystemandassumethatacustomerislostiftheclosestfacilityandallotherfacilitiesthathe/orshecanreacharefull.Bothpapers([ 13 14 ])obtainthesolutions,i.e.,thelocationofthefacilities,throughheuristicapproaches. Wenotethatthecapacitiesoffacilitiestobeopenedhasanimpactonboththetotalnumberandthelocationsoffacilities,particularlywhenthecongestionassociatedwiththepotentialfacilitiesistakenintoaccount.Inwhatfollows,wemodeleachfacilityasanM=M=c=cqueueingsystem,wherecisnumberofserversinthefacility,whichdesignatesthecapacityofthefacility.Thegoalofourmodelistoidentifylocationsoffacilitiesandspecifythecapacitylevelsofthefacilitiessimultaneously. Letfjbethexedcostofopeningafacilityatnodej2N,andcjkbetheoperatingcostofthefacilityjwithcapacitylevelk2K,whereK=f1;2;:::;KgisthesetofcapacitylevelsofanopenedER.Inthisproblem,thecapacityismeasuredasnumberofbedsinanER.Thatis,ifanERistobeopenedatcapacitylevelk2K,thentherearemkbedsintheER.WeassumethatthelengthsofstayofapatientintheERatnodejareexponentiallydistributedwithratej,andeachERismodeledasanM=M=c=c

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Wehavefoursetsofdecisionvariables.Therstsetispatientallocationvariablexij,whichtakesthevalueofoneifnodeiisservedbytheERservicefacilityatlocationj,zerootherwise.ThesecondsetisERlocationandcapacityallocationvariableyjk,whichtakesthevalueofoneifanERservicefacilityisplacedatnodejandoperatedatcapacitylevelk,zerootherwise.Thelasttwosetsarejandzj,whicharedeterminedoncethevaluesofxijandyjkareassigned. TheERSFLCPproblemcanbeformulatedasanonlinearintegerprogrammingformulationasfollows: minNXi=1NXj=1tnidijxij+NXj=1KXk=1(fj+cjk)yjk 98

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where(j;j;zj)istheprobabilityofdivertingpatientsfromanERservicefacilityatlocationj.Theformulasfor(j;j;zj)ofanM=M=c=cqueueingsystemcanbefoundinanyqueueingbook(e.g.,GrossandHarris,1998).Specically,(j;j;zj)canberepresentedasafunctionofarrivalratej,serviceratej,andbedcapacityzj,thatis(j;j;zj)=(j=j)zj Theobjectivefunction( 5{1 )minimizesthevalueoftimeofERpatientsandthecostsofopeningandoperatingERservicefacilities.Constraint( 5{2 )stipulatesthateachdemandnodemustbecoveredbyonefacility.Constraint( 5{3 )restrictsthatdemandnodescanonlybeassignedtoopenedfacilities.Constraint( 5{4 )imposestheupperboundonthenumberoffacilitiesopened.Constraint( 5{5 )statesthatanopenedfacilitymustbeassociatedwithasinglecapacitylevel.Constraints( 5{6 )and( 5{7 )obtainthearrivalratesandcapacitylevelsforallthefacilities.Constraint( 5{8 )imposestheupperboundontheprobabilityofallbedsbeingoccupiedforanopenedfacility.Finally,constraints( 5{9 )and( 5{10 )ensurethatdecisionvariablesarebinaryandnonnegativeintegers,respectively. Notethatthefactorialtermsinequation( 5{11 )maketheERSFLCPproblemintractable.Toovercomethisproblem,wereplacetheconstraint( 5{8 )byconstraint( 5{12 ), wherejkisthelargestvaluewhichsatisesinequality( 5{13 )andequation( 5{14 ).(jk;j;mk)=(jk=j)mk 99

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Asaresult,wetransformthepreviousnonlinearintegerprogrammingformulationtoalinearbinaryintegerprogrammingformulationasfollows: minNXi=1NXj=1tnidijxij+NXj=1KXk=1(fj+cjk)yjk 5-1 presentsthegeneralprocedureweusetoobtainasolutiontotheERSFLCPproblemusingaLagrangianrelaxationapproach.InFigure 5-1 ,nistheiterationcounter.Also,UBandLBdenotetheincumbentupperandlowerboundsontheobjectivefunctionvalueof 100

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endLagrangianRelaxation. Figure5-1. Pseudo-codeoftheLagrangianrelaxation 5{17 )and( 5{20 )withLagrangemultipliersand,respectively,whereandarematriceswithsizesNNandNK,respectively.TherelaxationyieldsthefollowingLagrangianproblem(ERSFLCP-LR): minNXi=1NXj=1(tnidij+ij+niXkjk)xij+NXj=1KXk=1(fj+cjkXiijjkjk+jkMj)yjkNXj=1KXk=1jkMj 101

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NotethattheLagrangianproblemcanbeseparatedintosubproblemsLXandLY,wheresubproblemLXcontainsvariablesxij,andthesubproblemLYcontainsvariablesyjkasfollows: (LX)minNXi=1NXj=1~dijxijsubjecttoNXj=1aijxij=18ixij2f0;1g8i;j(LY)minNXj=1KXk=1~cjkyjksubjecttoNXj=1KXk=1yjkpKXk=1yjk18jyjk2f0;1g8j;k Givenasetofand,bothsubproblemsLXandLYareeasytosolve.ForsubproblemLX,foreachi2Nthedecisionvariablexijissettooneif~dij~dijforallj2N.Similarly,subproblemLYcanbesolvedaccordingtoeachvariableyjk's 102

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5-2 depictsthepseudo-codeoftheheuristic. Theheuristicstartsbyresettingtheinfeasiblesolution(X,Y)accordingtothestrategyselectedrandomlyfromfollowing: 1.Foreachi,ifxij>0,thenPNk=1yjk>0;and 2.IfPNk=1yjk=0,thenxij=0foralli2N. Therststrategyresetsfacilityvariables,yjk,basedondemandallocation,xij,determinedfromsubproblemLX.Thefacilitiesareorderedaccordingtotheirpatientarrivalrates,i.e.,j=Pinixij,innon-increasingordern,andthentherstpfacilitiesaresettoopenatcapacitylevelone,i.e.,yj1=1.Thesecondstrategyresetsdemandallocationvariables,xij,accordingtosolutionyjkformsubproblemLY.Thatis,ifafacilityjisnotopened,thenthedemandnodesallocatedtofacilityjareresettonotcovered,i.e.,thecorrespondingvariablexij'sareresetto0andre-allocatethemtootherfacilitiesusingthenextprocedure. Line 2 resetsthecapacitylevelkoftheopenedfacilitiessuchthatthefacilitiesisopenedatappropriatecapacitylevelk,i.e.,Pinixijjk.Foranopenedfacilityj,ifthedemandallocatedexceedsitslargestcapacity,i.e.,Pinixij>jK,thenanallocateddemandnodeiisselectedrandomlyanditsxijisresetto0,untilPinixijjkholds. Next,thewhileloopinline 3 inFigure 5-2 ensuresthatalldemandnodesarecoveredandthenumberoffacilitiesopenedislessthanp.Thepseudo-codeofcoveringalldemandnodesispresentedinFigure 5-3 .Werandomlyselectanodeuwhichhasnotcoveredby 103

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Figure5-2. Pseudo-codeofthefeasiblesolutiongeneration anyfacility,andgenerateasetOpenFwhichcontainsthefacilitiesmeetingthefollowingcriteria: 1.Thefacilityjisopened,i.e.,Pkyjk>0; 2.Thefacilityjcancoveredthedemandnodeu,i.e.,auj=1;and 3.Afterallocatingnodeutothefacilityj,thesumofarrivalratesoftheallocateddemandnodedoesnotexceedfacilityj'smaximumcapacity,i.e.,Pinixij+nujK. IfthesetOpenFisempty,thenasetNotOpenFisgenerated.ThesetNotOpenFincludesthefacilitieswhicharenotopened,i.e.,Pkyjk=0,andthenodeuiswithintheircoveragerange,i.e.,auj=1forallj2N.Then,intheline 8 inFigure 5-3 weopenafacilityj2NotOpenFtocovernodeu.Therearemanystrategiesthatwecanapplytochoosewhichfacilitytoopen,suchas,openthefacilityj2NotOpenFwhichistheclosestonetonodeu,theonewiththelargestcapacity,ortheonewiththesmallestxcost.Hereweapplythethreestrategiesandndthecorrespondingfacilitiesforeachofthem.Ifthesestrategiesyielddierentfacilityoptions,werandomlyselectoneofthemtoopen.Oncealldemandnodesarecoveredbytheopenedfacilities,thecapacitylevelofeachopenedfacilityisresettoappropriatelevelk2K(line 14 inFigure 5-3 )suchthatPinixijjk. Sofar,wehaveensuredthatalldemandnodesarecoveredbyfacilitieswhichareopened,andthedemandallocationdoesnotexceedthemaximumcapacityoftheopened 104

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Figure5-3. Pseudo-codeofcoveringalldemandnodes facilities.Thenextthingtodoistoinspectwhetherthenumberoffacilitiesopenedisnolargerthanp.Ifthenumberoffacilitiesopenedislessthanorequaltop,thenafeasiblesolutionisgenerated.Otherwise,oneoftheopenedfacilitiesisselectedrandomlyandclosed,untilthenumberoffacilitiesopenedisnolargerthanp.Inaddition,theassociateddemandnodesareresettonotcovered,i.e.,resetxij=0,wherefacilityjischosentoclose. Figure5-4. Pseudo-codeofclosingfacilities 105

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5{17 )and( 5{20 )by0xij1and0yjk1,respectively.Then,CPLEXisusedtosolvetheERSFLCPrproblem,andthedualinformationoftheconstraintsets0xij1and0yjk1areextractedtosettheinitialvaluesofLagrangianmultipliers.Letrandrbethedualvaluesassociatedwiththeconstraintsets0xij1and0yjk1,respectively.Themultipliers0and0ofERSFLCP-LRareinitializedbytheequations0=rand0=r. WethenapplythemethoddescribedbyFisher(1981)toupdatethemultipliers.Ateachiterationn,thestepsizetnisobtainedbysn=Bn(^n) 106

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5-1 ,andtheotherthreeparametersaresetatlevel2. Table5-1. Experimentalfactorsettings ParametersLevel1Level2Level3 Experiment1p51020Experiment2K()5(8)8(5)10(4)Experiment30.5%1%2%Experiment4t($/perminute)2550100 Foreachexperimentalsetting,wegenerate30randominstances.Eachrandominstanceisobtainedbygeneratingparametersthatcorrespondtopatientarrivalrates,servicerates,xedandoperatingcostsanddistancebetweeneachnodepair.Specically,eachparameterisobtainedbyusingtheformula:u(meanvalueoftheparameter),whereuisdrawnfromthedistributionU[0:5;1:5].Themeanvaluesoftherandomparametersarelistedasfollows: 107

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Last,thecoveragerange,d,issetto50minutesforallinstances. Foreachinstance,theLagrangianrelaxationalgorithmdescribedinSection 5.4 isapplied.TheupperlimitofthenumberofLRiterations(nmax)issetat100,000andtheupperlimitofthenumberofconsecutiveiterationsfailtoimprovethebestknownfeasiblesolution(umax)issetat10,000.TheparameterBusedtomodiedthestepsizeisinitializedat2,anddividedby1.5ifthelowerboundisnotimprovedfor3Niterations. 5-5 depicttheupperandlowerbounds,respectively,obtainedfromtheLRiterationswiththedualinformationofERSFLCPr.ThedashlinesshowtheLRresultswithoutapplyingthedualinformationofERSFLCPr.ThedualinformationofERSFLCPrprovidesagoodlowerboundsolutionwhichguidestheheuristictondthebestupperboundsolutionearlierthantheguidanceofthelowerboundobtainedfromusingtheLagrangianmultipliersgeneratedfromscratch.TheconvergencegapsinFigure 5-5 aregivenby^~ where^and~aretheupperandlowerboundsofoptimalobjectivevalueofERSFLCP-LRobtainedfromtheLRalgorithm. Table 5-2 toTable 5-5 reporttheresultsoffourexperiments.Ingeneral,LRperformsquitewellinsolvingtheERSFLCPproblemintermsoftheconvergencegapandthecomputationaleort(i.e.,CPUtime)required.Forthenetworkwith25nodes,LRtakesless2secondstoobtainthesolutionswithconvergencegapslessthan5%onaverage.Forthelargestnetworktested,i.e.,100nodes,theaverageCPUtimeisincreased,althoughitisstilllessthan60seconds.Moreover,LRdoesnotappeartoexperiencesignicantincreaseinconvergencegapduetoanincreaseinproblemsize. 108

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ConvergencespeedofthemodiedLR Therstexperimentshowsthatthesmallerthevalueofpis,thehighertheCPUtimerequiredtoobtainthesolution,sincesmallerpresultsinatighterconstraintonnumberoffacilitiesopened.Inparticular,thistrendcanbeobservedintheproblemswithsmall-andmedium-sizenetworks.Nevertheless,theproblemcanstillbesolvedwithin40secondsevenforlarge-sizenetworks. Table5-2. Experiment1:eectsofmaximumnumberoffacilitiesopened Networksize2550100Average GapTimeGapTimeGapTimeGapTimep(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.) 53.01.14.44.95.841.84.415.9103.10.93.04.33.835.53.313.6202.50.93.33.22.929.42.911.2 Average2.91.03.64.14.235.63.513.6 Inthesecondexperiment,weassumethatatmost40bedsareusedforanopenedER,sotheincrementalamountofbedspercapacitylevelarechangedwiththesettingofK.AlargeKvaluerepresentsthatcapacityisincreasedatsmallscale,i.e.,smallbatchsize,andincreasesthenumberofvariablesoftheunderlyingformulationthetheERSFLCPproblem.Asaresult,theCPUtimeisincreasedasKincreases,while 109

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Table5-3. Experiment2:eectsofcapacitysetting Networksize2550100Average GapTimeGapTimeGapTimeGapTimeK(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.) 5(8)0.70.82.24.23.631.42.112.18(5)3.21.12.94.13.934.03.313.110(4)3.61.13.34.53.935.93.613.8 Average2.51.02.84.33.833.83.013.0 ThethirdexperimentinvestigatestheimpactofthediversionprobabilityontheperformanceoftheproposedLRapproach.Giventhecapacitylevelk,thesmallerincursthesmallervalueofthemaximumarrivalratesthatafacilityisabletoserve,i.e.,jkinconstraintset( 5{20 ).Thatis,theconstraintset( 5{20 )becomestighterasdecreases.AsshowninTable 5-4 ,theCPUtimeslightlyincreasesasdecreasesfrom2.0%to1.0%.However,whenthevalueofisfurtherreducedto0.5%,therequiredCPUtimedecreases,whichisdierentfromthetrendthatweobservebefore.Theconvergencegapdoesnotappeartobeaectedbysignicantly;inaverage,thegapsare3.6%and2.9%when'sare0.5%and2.0%,respectivley. Table5-4. Experiment3:eectsofdiversionprobability Networksize2550100Average GapTimeGapTimeGapTimeGapTime(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.) 0.5%3.70.93.34.13.834.93.613.31.0%3.11.03.04.33.956.63.320.62.0%2.40.92.73.63.548.32.917.6 Average3.10.93.04.03.746.63.317.2 Thelastexperimentexplorestheeectsofpatients'valueoftimetontheperformanceoftheproposedLRapproach.Table 5-5 showsthatthehigherthepatients'timevalueis,thesmallertheconvergencegapis.WealsoobservethattheCPUtimeisnotsignicantlyaectedbythechangeoft. 110

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Experiment4:eectsoftimevalue Networksize2550100Average GapTimeGapTimeGapTimeGapTimet(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.) 254.91.14.95.85.645.85.117.6503.01.12.94.73.849.03.218.21001.81.11.75.42.750.82.119.1 Average3.21.13.25.34.048.53.518.3 Toillustratetheperformanceoftheheuristicdevelopedforobtainingtheupperbound(feasible)solutionateachiteration,weuseCPLEXtosolvetheERSFLCPproblem.Theparameters,p,K,andtaresettotheirrespectivevaluesforlevel2inTable 5-1 .Foreachtestinstance,CPLEXisstoppediftherelativestoppingtoleranceof0.01%issatised,orCPUtimeof3,600secondsisused.Table 5-6 summarizestheresults,wheretherelativeerrorofeachinstanceisgivenbyObjofCPLEX-UpperboundofLR ObjofCPLEX100%: 5-6 showsthatthedevelopedheuristicisveryeective.Forsmall-andmedian-sizeproblems,theupperboundsobtainedfromtheheuristicareveryclosetotheoptimalobjectivevalues;theaverageandmaximumrelativeerrorsarelessthan0.3%and1.5%,respectively.Fortheproblemwithlargestsize,CPLEXisnotabletondtheoptimalsolutionwithin3,600seconds.Therefore,weusethebestresultsthatCPLEXcanndwithin3,600secondstocomparewiththeresultsobtainedbyheuristic.Thecomparisonfurtherensurestheeectivenessoftheheuristicdevelopedhere.Theheuristicnotonlyspendsmuchlesstime(36secondsinaverage)thanCPLEX,butalsondsolutionswithbetterqualityin50%oftheinstances. 111

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Performanceoftheheuristic CPLEXHeuristicErr.CPLEXHeuristicErr.CPLEXHeuristicErr.Instance(sec.)(sec.)(%)(sec.)(sec.)(%)(sec.)(sec.)(%) 10.40.00.831.83.30.1>360026.60.420.70.01.484.34.50.1>360059.7-1.330.30.01.144.93.50.0>360026.00.440.50.01.1109.94.70.2>360039.70.250.80.00.882.45.10.5>360031.90.260.70.00.847.97.50.9>360033.30.170.20.00.9115.03.50.1>360029.4-0.580.20.00.818.74.10.0>360030.50.190.90.01.027.73.50.2>360051.4-0.4100.20.00.932.64.60.2>360027.80.7110.80.11.1130.95.10.5>360065.60.1120.30.00.965.25.30.2>360030.1-0.3130.60.00.965.53.00.1>360022.8-1.8140.30.00.939.94.80.6>360039.5-0.4150.60.00.925.63.40.3>360025.9-0.3160.50.00.9456.43.70.3>360055.3-0.8170.60.01.047.94.30.6>360034.80.4180.80.01.634.73.20.1>360041.50.2190.70.00.847.33.50.3>360019.50.2200.80.00.8119.83.81.2>360024.4-0.2210.40.00.853.53.80.1>360025.70.2220.40.00.846.73.40.0>360025.4-1.2230.20.00.817.47.50.5>360033.70.3240.30.00.862.83.30.1>360049.6-1.1250.50.00.9502.13.50.2>360028.10.5260.80.00.953.63.60.3>360023.6-1.4270.60.01.027.43.70.4>360019.9-0.8280.60.00.9146.64.80.3>360037.10.2290.20.00.939.95.81.2>360060.2-1.2300.10.00.9176.94.10.0>360046.2-0.2 Min.0.10.00.817.43.00.0>360019.5-1.8Avg.0.50.00.991.84.30.3>360035.5-0.3Max.0.90.11.6502.17.51.2>360065.60.7

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Animmediateextensionofourmodelistoincludetheclosest-assignmentconstraintswhichensurethateachdemandpointisallocatedtotheclosestopenfacility.Someotherpracticallyrelevantvariationsareconcernedwiththealternativeobjectivefunctions,suchasprotmaximization,andtravelandservicetimesminimization.AnothercloselyrelatedproblemisthatgivensomeERservicefacilitiesareopenedalreadyandBnewbedsareconsideredtobeaddedatexistingfacilitiesornewlyopenedfacilities.Wenotethatthiscanbedenedbyaddingtheconstraintsoncapacityavailable,distinguishingthesetofexistingandnewERlocations,andreplacingthexedcosttocapacityexpansioncostofexistingfacilities. 113

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Inthiswork,wehavepresentedtheintegrateduseofoptimizationandqueueingtheorytodeterminetheoptimalcapacityplanforhealthcaresystems. Chapter 2 detailedtheaggregatehospitalbedcapacityplanning(AHBCP)problemandanetworkowapproachtospecifytheoptimalbedcapacityplanningdecisionsthroughoutaniteplanninghorizonforhospitals.Inthischapter,ahospitalwasmodeledasaG=G=cqueueingsystemwithasinglebedtypeandasinglepatientclass.Wedemonstratedthatforrealistic-sizedcapacityplanningproblems,ournetworkformulationisnotcomputationallyintensive,andallowsustoobtainoptimalbedcapacityplansquickly. Chapter 3 introducedthehealthcareteamcapacityplanning(HCTCP)problem,inwhichtheunderlyingqueueingsystemwasmorecomplexthantheoneusedforAHBCP.Inparticular,weconsideredaqueueingsystemwheretherearetwoclassesofpatientsandtwotypesofcareteams,whereserviceratesarepatient-classdependentandonetypeofcareteamcansubstitutefortheother.Wedevelopedqueueingmodelsforbothpreemptiveandnon-preemptivecases,anddevelopedapproximationprocedurestoestimatetheaveragetimethateachpatientclassspendsinthesystem.Theresultsfromapproximationmethodwerethenincorporatedintotheoptimizationmodeltodeterminetheminimalcostcapacityplanofhealthcareteamsthroughoutaniteplanninghorizon.Ourcomputationalstudyshowedthatourapproximationapproachprovidessucientlyaccurateresultsthatcanbeusedinpracticetomakelong-termhealthcareteamservicecapacityplanningdecisions. AftertheaggregatebedcapacitywasspeciedinChapter 2 ,wedevelopedthehospitalbedallocation(HBA)modeltoobtainthebalancedbedallocationamongdierentmedicalcareunits,whichweremodeledasM=M=c=cqueueingsystems.Toecientlysolvetheproblem,weproposedthreesolutionapproachesincluding 114

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Chapter 5 developedaemergencyroomservicesfacilitylocationandcapacityplanning(ERSFLCP)model,whereeachemergencyroomservicefacilityisviewedasaM=M=c=cqueueingsystem.Themodelwasdesignedtosimultaneouslydeterminethenumberoffacilitiesopenedandtheirrespectivelocationsaswellasthecapacitylevelsofthefacilities(captureintermsofnumberofbeds)sothattheprobabilitythatthefacilityisfullandincomingpatientshavetobediverted(i.e.,diversionprobability)isnotlargerthanaparticularthreshold.ALagrangianrelaxationapproachwasproposedtoobtainfacilitylocationsandcapacityplan.TheexperimentalresultsillustratedthattheLagrangianrelaxationalgorithmisveryecientinsolvingtheproblem,andthedevelopedheuristicprovidessolutionswithgoodquality. Inourworktodate,weprimarilyfocusedonstrategiclevel,long-termplanningdecisionswherewemadesomesimplifyingassumptionsastothecapabilitiesoftheresourcesandthearrivalratesandlengthsofstayforthepatients.Specically,weassumedthattheavailablebedsinaservice(inChapters 3 4 ,and 5 )orthehospital(inChapter 2 )areidentical.Similarly,weassumedthatthearrivalratesandlengthofstayforthepatientsareidentical(inChapters 2 and 5 ).Butwealsoconsideredthecasewherepatientscanbegroupedintotwoclasses(eachofwhichcorrespondstoanacuitylevel)orintomultipleclasses(eachofwhichcorrespondstoaparticularspeciality)tomodelarrivalratesandlengthsofstay.Aswemainlyfocusedonstrategicleveldecisionmaking,theseassumptionsarejustiable.However,formoredetailedplanningthereisaneedtotakeotherrealisticconsiderationsintoaccount. Therearetypicallymultipletypesofbeds(e.g.,adultintensivecarebeds,pediatricintensivecarebeds,burnintensivecarebeds,medical/surgicalbeds),i.e.,multiple 115

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Inourwork,wehaveprimarilyfocusedontheobjectiveofminimizingtotalcosts(inChapters 2 3 ,and 5 )andbalancingworkloadacrossunits(inChapter 4 ).Nowadays,hospitalsarefocusingonrevenuemanagementpracticestoimprovetheirnancialsituation.Thismodernrevenuemanagementculturerequireshospitaladministratorstofocusonmaximizingprotratherthanonminimizingoperatingcosts.Therefore,futureworkshouldexaminetherevenueaspectsofhospitaloperationsandfocusonprotmaximizationtypeobjectives.However,thisshiftinemphasisfromcostminimizationtoprotmaximizationshouldnotignorethequalityaspectsassociatedwiththedeliveryofhealthcareservices.Inourresearch,wefocusedontimeliness(e.g.,averageservicetime,averagewaitingtime)andaccess(e.g.,diversionprobability)toquantifyservicequality.Futureresearchintheareashouldconcentrateondevelopingothermetricstomodelotheraspectsofservicequality,suchaspatientsafetyandserviceeectiveness. 116

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[1] AgencyforHealthcareResearchandQuality.Lasttimeaccessed:November2007,http://www.ahrq.gov/research/havbed/definitions.htm. [2] R.K.Ahuja,T.L.Magnanti,J.B.Orlin,NetworkFlows:Theory,Algorithms,andApplications,PrenticeHall,NewJersey,1993. [3] L.A.Aiken,S.P.Clarke,D.M.Sloaneetal.,Nursesreportsonhospitalcareinvecountries,HealthAairs20(2001)43-53. [4] L.A.Aiken,S.P.Clarke,D.M.Sloaneetal.,Hospitalnursestangandpatientmortality,nurseburnout,andjobdissatisfaction,JAMA288(2002)1987{1993. [5] E.Akcal,M.J.C^ote,C.-ILin,Anetworkowapproachtooptimizinghospitalbedcapacitydecisions,HealthCareManagementScience9(2006)391{404. [6] R.Akkerman,M.Knip,Reallocationofbedstoreducewaitingtimeforcardiacsurgery,HealthCareManagementScience7(2004)119{126. [7] AmericanHospitalAssociation,HospitalStatistics2005Edition,HealthForum,Chicago,2005. [8] P.-A.Andersson,E.Varde,F.Diderichsen,ModellingofresourceallocationtohealthcareauthoritiesinStockholmCounty,HealthCareManagementScience3(2000)141{149. [9] R.Batta,O.Berman,Q.Wang,Stangandswitchingcostsinaservicecenterwithexibleservers,EuropeanJournalofOperationalResearch177(2007)924{938. [10] G.J.Bazzoli,L.R.Brewster,G.Liu,S.Kuo,DoesU.S.hospitalcapacityneedtobeexpanded?HealthAairs22(2003)40-54. [11] D.Bellandi,Runningatcapacity,ModernHealthcare110(1999)112-113. [12] P.Beraldi,M.E.Bruni,D.Conforti,Designingrobustemergencymedicalserviceviastochasticprogramming,EuropeanJournalofOperationalResearch158(2004)183{193. [13] O.Berman,D.Krass,J.Wang,LocatingServiceFacilitiestoReduceLossDemand,IIETransactions38(2006)933{946. [14] O.Berman,R.Huang,S.Kim,S.,M.Menezes,Locatingcapacitatedfacilitiestomaximizecaptureddemand,IIETransactions39(2007)1015{1029. [15] D.M.Berwick,Wecancutcostsandimprovecareatthesametime,MedicalEconomics180(1996)185-187. [16] G.R.Bitran,D.Tirupati,Capacityplanninginmanufacturingnetworkswithdiscreteoptions,AnnalsofOperationsResearch17(1989)119{136. 117

PAGE 118

G.R.Bitran,D.Tirupati,Tradeocurves,targetingandbalancinginmanufacturingqueueingnetworks,OperationsResearch37(1989)547{564. [18] J.T.Blake,M.W.Carter,Agoalprogrammingapproachtostrategicresourceallocationinacutecarehospitals,EuropeanJournalofOperationalResearch140(2002)541{561. [19] J.P.C.Blanc,Performanceevaluationofpollingsystemsbymeansofthepower-seriesalgorithm,AnnalsofOperationsResearch35(1992)155{186. [20] J.P.C.Blanc,Performanceanalysisandoptimizationwiththepower-seriesalgorithm,inL.Donatiello,R.Nelson(eds),PerformanceanalysisandoptimizationwithpowerseriesalgorithmSpringerVerlag,Berlin,53{80,1993. [21] B.Boey,D.Yates,R.D.Galv~ao,AnalgorithmtolocateperinatalfacilitiesinthemunicipalityofRiodeJaneiro,JournaloftheOperationalResearchSociety54(2003)21{31. [22] J.Bowers,B.Lyons,G.Mould,T.Symonds,Modellingoutpatientcapacityforadiagnosisandtreatmentcentre,HealthCareManagementScience8(2005)205{211. [23] J.Bowers,G.Mould,Managinguncertaintyinorthopaedictraumatheatres,EuropeanJournalofOperationalResearch154(2004)599{608. [24] J.Bowers,G.Mould,Ambulatorycareandorthopaediccapacityplanning,HealthCareManagementScience8(2005)41{47. [25] M.L.Brandeau,F.Sainfort,W.P.Pierskalla,OperationsResearchandHealthCare:AHandbookofMethodsandApplications,Springer,Boston,2004. [26] K.M.Bretthauer,M.J.C^ote,Amodelforplanningresourcerequirementsinhealthcareorganizations,DecisionSciences29(1998)243{270. [27] M.J.Brusco,M.J.Showalter,Constrainednursestaanalysis,OMEGA:InternationalJournalofManagementScience21(1993)175{186. [28] E.K.Burke,P.DeCausmaecker,G.VandenBerghe,H.VanLandeghem,Thestateoftheartofnurserostering,JournalofScheduling7(2004)441{499. [29] F.Cerne,J.Montague,Capacitycrisis,HospitalsandHealthNetworks68(19)(1994)30{36. [30] J.K.Cochran,A.Bharti,Stochasticbedbalancingofanobstetricshospital,HealthCareManagementScience9(2006)31{45. [31] R.C.CoileJr.,Futurescan2002:aforecastofhealthcaretrends20022006,HealthAdministration,Chicago,2002. [32] M.J.C^ote,S.L.Tucker,Fourmethodologiestoimprovehealthcaredemandforecasting,HealthcareFinancialManagement55(2001)54-58. 118

PAGE 119

R.Davies,Simulationforplanningservicesforpatientswithcoronaryarterydisease,EuropeanJournalofOperationalResearch72(1994)323{332. [34] K.Davis,S.C.Schoenbaum,A.-M.Audet,A2020VisionofPatient-CenteredPrimaryCare,JournalofGeneralInternalMedicine20(2005)953{957. [35] L.Delesie,A.Kastelein,F.Merode,J.M.H.Vissers,Managinghealthcareunderresourceconstraints,EuropeanJournalofOperationalResearch105(1998)247{247. [36] T.A.Feo,M.G.C.Resende,Aprobabilisticheuristicforacomputationallydicultsetcoveringproblem,OperationsResearchLetters8(1989)67-71. [37] M.L.Fisher,TheLagrangianrelaxationmethodforsolvingintegerprogrammingproblems,ManagementScience27(1981)1{18. [38] S.Flessa,Whereeciencysaveslives:Alinearprogrammefortheoptimalallocationofhealthcareresourcesindevelopingcountries,HealthCareManagementScience3(2000)249{267. [39] R.D.Galv~ao,L.G.A.Espejo,B.Boey,AhierarchicalmodelforthelocationofperinatalfacilitiesinthemunicipalityofRiodeJaneiro,EuropeanJournalofOperationalResearch138(2002)495{517. [40] R.D.Galv~ao,L.G.A.Espejo,B.Boey,D.Yates,Loadbalancingandcapacityconstraintsinahierarchicallocationmodel,EuropeanJournalofOperationalResearch172(2006)631{646. [41] D.E.Goldberg,GeneticAlgorithmsinSearch,OptimizationandMachineLearning,Addison-Wesley,Wokingham,England,1989. [42] S.M.Goldstein,P.T.Ward,G.K.Leong,T.W.Butler,Theeectoflocation,strategy,andoperationstechnologyonhospitalperformance,JournalofOperationsManagement20(2002)63{75. [43] F.Gorunescu,S.I.McClean,P.H.Millard,Usingaqueueingmodeltohelpplanbedallocationinadepartmentofgeriatricmedicine,HealthCareManagementScience5(2002)307{312. [44] L.Green,Aqueuingsystemwithgeneral-useandlimited-useservers,OperationsResearch33(1985)168{182. [45] L.V.Green,Howmanyhospitalbeds?Inquiry38(2002/2003)400-412. [46] L.V.Green,V.Nguyen,Strategiesforcuttinghospitalbeds:theimpactonpatientservice,HealthServicesResearch36(2001)421{442. [47] S.Groothuis,A.Hasman,P.E.J.vanPoletal.,Predictingcapacitiesrequiredincardiologyunitsforheartfailurepatientsviasimulation,ComputermethodsandProgramsinBiomedicine74(2)(2004)129{141. 119

PAGE 120

D.Gross,C.M.Harris,FundamentalsofQueueingTheory(3nded.),JohnWileyandSons,NewYork,1998. [49] K.Grumbach,T.Bodenheimer,Canhealthcareteamsimproveprimarycarepractice?JAMA:thejournaloftheAmericanMedicalAssociation291(2004)1246{1251. [50] C.Hayward,Whatarewegoingtodowithallourexcesscapacity?HealthCareStrategicManagement16(1998)20{23. [51] P.R.Harper,Aframeworkforoperationalmodellingofhospitalresources,HealthCareManagementScience5(2002)165{173. [52] G.W.Harrison,A.Shafer,M.Mackay,ModellingVariabilityinHospitalBedOccupancy,HealthCareManagementScience8(2005)325{334. [53] J.H.Holland,AdaptationinNaturalandArticialSystems,UniversityofMichiganPress,AnnArbor,1975. [54] G.Hooghiemstra,M.Keane,S.V.D.Ree,Powerseriesforstationarydistributionsofcoupledprocessormodels,SIAMJournalonAppliedMathematics48(1988)1159{1166. [55] InstituteofMedicine,CrossingtheQualityChasm,NationalAcademyPress,Washington,D.C,2001. [56] H.Jia,F.Ordonez,M.M.Dessouky,Amodelingframeworkforfacilitylocationofmedicalservicesforlarge-scaleemergencies,IIETransactions39(2007)41{55. [57] A.M.Johnson,Capacityplanningforthefuture,JournalofHealthCareFinance24(1997)72-75. [58] J.B.Jun,S.H.Jacobson,J.R.Swisher,Applicationofdiscrete-eventsimulationinhealthcareclinics:Asurvey,JournaloftheOperationalResearchSociety50(2)(1999)109{123. [59] E.P.C.Kao,C.Lin,TheM/M/1queuewithrandomlyvaryingarrivalandservicerates:aphasesubstitutionsolution,ManagementScience35(1989)561{570. [60] E.P.C.Kao,K.S.Narayanan,Computingsteady-stateprobabilitiesofanonpreemptiveprioritymultiserverqueue,ORSAJournalonComputing2(1990)211{218. [61] E.P.C.Kao,M.Queyranne,Budgetingcostsofnursinginahospital,ManagementScience31(1985)608{621. [62] E.P.C.Kao,G.G.Tung,Aggregatenursingrequirementplanninginapublichealthcaredeliverysystem,Socio-EconomicsPlanningScience15(1981)119{127. 120

PAGE 121

E.P.C.Kao,G.G.Tung,Bedallocationinapublichealthcaredeliverysystem,ManagementScience27(1981)507{520. [64] E.P.C.Kao,S.D.Wilson,Analysisofnonpreemptivepriorityqueueswithmultipleserversandtwopriorityclasses,EuropeanJournalofOperationalResearch118(1999)181{193. [65] E.H.Kaplan,M.Johri,Treatmentondemand:anoperationalmodel,HealthCareManagementScience3(2000)171{183. [66] S.Kavanagh,J.Cowan,Reducingriskinhealthcareteams:anoverview,ClinicalGovernance9(2004)200-204. [67] S.Kim,I.Horowitz,K.K.Young,T.A.Buckley,Analysisofcapacitymanagementoftheintensivecareunitinahospital,EuropeanJournalofOperationalResearch115(1999)36{46. [68] S.Kim,I.Horowitz,K.K.Young,T.A.Buckley,Flexiblebedallocationandperformanceintheintensivecareunit,JournalofOperationsManagement18(2000)427{443. [69] L.Kleinrock,Queueingsystems:TheoryVol.1,Wiley,NewYork,1975. [70] N.Koizumi,E.Kuno,T.E.Smith,Modelingpatientowsusingaqueuingnetworkwithblocking,HealthCareManagementScience8(2005)49{60. [71] G.Koole,OntheuseofthepowerseriesalgorithmforgeneralMarkovprocesses,withanapplicationtoaPetrinet,INFORMSJournalonComputing9(1997)51{56. [72] D.H.Kropp,R.C.Carlson,Recursivemodelingofambulatoryhealthcaresettings,JournalofMedicalSystems1(1977)123{135. [73] R.J.Kusters,P.M.A.Groot,Modellingresourceavailabilityingeneralhospitalsdesignandimplementationofadecisionsupportmodel,EuropeanJournalofOperationalResearch88(1996)428{445. [74] G.Latouche,V.Ramaswami,Alogarithmicreductionalgorithmforquasi-birth-and-deathprocesses,JournalofAppliedProbability30(1993)650{674. [75] G.Latouche,V.Ramaswami,IntroductiontoMatrixAnalyticMethods,SIAM,Philadelphia,PA,1999. [76] L.X.Li,W.C.Benton,Performancemeasurementcriteriainhealthcareorganizations:Reviewandfutureresearchdirections,EuropeanJournalofOperationalResearch93(1996)449{468. [77] L.X.Li,W.C.Benton,G.K.Leong,Theimpactofstrategicoperationsmanagementdecisionsoncommunityhospitalperformance,JournalofOperationsManagement20(2002)389{408. 121

PAGE 122

E.Litvak,P.I.Buerhaus,F.Davido,F.etal.,Managingunnecessaryvariabilityinpatientdemandtoreducenursingstressandimprovepatientsafety,JointCommissionJournalonQualityandPatientSafety31(2005)330-338. [79] M.Mackay,Practicalexperiencewithbedoccupancymanagementandplanningsystems:anAustralianview,HealthCareManagementScience4(2001)47{56. [80] M.Mackay,M.Lee,ChoiceofModelsfortheAnalysisandForecastingofHospitalBeds,HealthCareManagementScience8(2005)221{230. [81] V.Marianov,C.ReVelle,Thequeueingmaximalavailabilitylocationproblem:Amodelforthesitingofemergencyvehicles,EuropeanJournalofOperationalResearch93(1996)110{120. [82] A.H.Marshall,S.I.McClean,C.M.Shapcott,P.H.Millard,Modellingpatientdurationofstaytofacilitateresourcemanagementofgeriatrichospitals,HealthCareManagementScience5(2002)313{319. [83] A.H.Marshall,C.Vasilakis,E.El-Darzi,Lengthofstay-basedpatientowmodels:recentdevelopmentsandfuturedirections,HealthCareManagementScience8(2005)213{220. [84] B.J.Masterson,T.G.Mihara,G.Miller,S.C.Randolph,M.E.Forkner,A.L.Crouter,Usingmodelsanddatatosupportoptimizationofthemilitaryhealthsystem:Acasestudyinanintensivecareunit,HealthCareManagementScience7(2004)217{224. [85] W.E.McAleer,I.A.Naqvi,Therelocationofambulancestations:Asuccessfulcasestudy,EuropeanJournalofOperationalResearch75(1994)582{588. [86] L.W.Morton,N.Bills,D.Kay,Boostinglocaleconomiesthroughhealthcareeconomicdevelopment,Lasttimeaccessed:August2005,http://www.cardi.cornell.edu/economic development/community economic renewal/000274.phpCommunityandRuralDevelopmentInstitute,CornellUniversity. [87] M.F.Neuts,Matrix-GeometricSolutionsinStochasticModels,JohnsHopkinsUniversityPress,Maryland,1981. [88] J.M.Nguyen,P.Six,T.Chaussalet,D.Antonioli,P.Lombrail,P.LeBeux,Anobjectivemethodforbedcapacityplanninginahospitaldepartment-Acomparisonwithtargetratiomethods,Methodsofinformationinmedicine46(4)(2007)399{405. [89] W.P.Pierskalla,Healthcaredelivery,PresentedattheNationalScienceFoundationWorkshoponEngineeringtheServiceSector,Atlanta,2001. 122

PAGE 123

W.P.Pierskalla,D.Brailer,Applicationsofoperationsresearchinhealthcaredelivery,inBeyondtheprotmotive:publicsectorapplicationsandmethodology,HandbooksinOR&MS,S.Pollock,A.Barnett,M.Rothkopf(eds),Vol6,North-Holland,NewYork,1994. [91] W.P.Pierskalla,D.Wilson,Reviewofoperationsresearchimprovementsinpatientcaredeliverysystems,Workingpaper,UniversityofPennsylvania,Philadelphia,1989. [92] S.Rahman,D.K.Smith,Useoflocation-allocationmodelsinhealthservicedevelopmentplanningindevelopingnations,EuropeanJournalofOperationalResearch123(2000)437{452. [93] J.F.Repede,J.J.Bernardo,DevelopingandvalidatingadecisionsupportsystemforlocatingemergencymedicalvehiclesinLouisville,Kentucky,EuropeanJournalofOperationalResearch75(1994)567{581. [94] J.C.Ridge,S.K.Jones,M.S.Nielsen,A.K.Shahani,Capacityplanningforintensivecareunits,EuropeanJournalofOperationalResearch105(1998)346{355. [95] S.M.Ryan,Capacityexpansionforrandomexponentialdemandgrowthwithleadtimes,ManagementScience50(2004)740{748. [96] F.Sainfort,WhereisOR/MSinthepresentcrisesinhealthcaredelivery?PresentedattheInstituteforOperationsResearchandtheManagementSciencesAnnualMeeting,Miami,2001. [97] D.P.Schneider,K.E.Kilpatrick,AnOptimumManpowerUtilizationModelforHealthMaintenanceOrganizations,OperationsResearch23(1975)869{889. [98] L.Shi,D.A.Singh,EssentialsoftheUSHealthCareSystem,JonesandBartlettPublishers,Maryland,2005. [99] R.A.Shumsky,Approximationandanalysisofaqueueingsystemwithexibleandspecializedservers,ORSpectrum,SpecialIssueonCallCenterManagement26(3)(2004)307{330. [100] D.Sinreich,Y.Marmor,Emergencydepartmentoperations:thebasisfordevelopingasimulationtool,IIETransactions37(3)(2005)233{245. [101] V.L.Smith-Daniels,S.B.Schweikhart,D.E.Smith-Daniels,Capacitymanagementinhealthcareservices:reviewandfutureresearchdirections,DecisionScience19(1988)899-918 [102] C.Smith,C.Cowan,A.Sensenig,A.Catlin,HealthSpendingGrowthSlowsin2004,HealthAairs25(2006)186{196. [103] L.Snyder,Facilitylocationunderuncertainty:areview,IIETransactions38(2006)537{554. 123

PAGE 124

L.Snyder,M.Daskin,ReliabilityModelsforFacilityLocation:TheExpectedFailureCostCase,TransportationScience39(3)(2005)400{416. [105] D.A.Stanford,W.K.Grassmann,Thebilingualserversystem:aqueueingmodelfeaturingfullyandpartiallyqualiedservers,INFOR31(4)(1993)261{277. [106] J.Thorson,Lasttimeaccessed:August2006,http://sepwww.stanford.edu/oldreports/sep20/20 11 abs.html. [107] UnitedStatesDepartmentofLaborBureauofLaborStatistics,Lasttimeaccessed:August2006,http://www.bls.gov/oco/cg/cgs035.htm#emply. [108] M.Utley,S.Gallivan,K.Davis,P.Daniel,P.Reeves,J.Worrall,Estimatingbedrequirementsforanintermediatecarefacility,EuropeanJournalofOperationalResearch150(2003)92{100. [109] A.H.vanZon,G.J.Kommer,Patientowsandoptimalhealth-careresourceallocationatthemacro-level:adynamiclinearprogrammingapproach,HealthCareManagementScience2(1999)87{96. [110] J.M.H.Vissers,Patientow-basedallocationofinpatientresources:Acasestudy,EuropeanJournalofOperationalResearch105(1998)356{370. [111] T.D.Vries,R.E.Beekman,Applyingsimpledynamicmodellingfordecisionsupportinplanningregionalhealthcare,EuropeanJournalofOperationalResearch105(1998)277{284. [112] S.Yan,ApproximatingReducedCostsunderDegeneracyinaNetworkFlowProblemwithSideConstraints,Networks27(1996)267{278. 124

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Chin-ILinwasborninTaipei,Taiwan.ShereceivedherB.S.andM.SincivilengineeringfromtheNationalCentralUniversityinTaiwanin1994and1996,respectively.From1997to2002,sheworkedforChinaAirlines,andhermajortasksincludeddemandforecast,marketanalysis,routeanalysis,andeetplanning.ShepursuedhermasteranddoctoraldegreesintheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridasince2002.Chin-I'smainresearchinterestisOperationsResearch,andtopicsofspecialinterestarehealthcaremanagementandairlineight/crewscheduling.Thusfar,herresearchhasfocusedoncapacitymanagementinhealthcaredelivery. 125